I know one shots are shorter and only last one session (hence the name) compared to a campaign where you play through an ongoing story. But besides that, what are the advantages and disadvantages of running one shots compared to campaigns?

For now at least, I don't see any reason for playing a one shot over a campaign. You take the time to create a character and end up using it one time, not feeling any attachment to your, or fellow player's, characters; and thus, in my eyes at least, would not enjoy as much compared to a campaign.

What are the pros and cons of playing a one shot instead of a campaign?

On definitions

Technically, one-shot can mean different things to different people. It doesn't always literally mean a story that can be wrapped up in one session (though that is usually the goal). However, it does always refer to an adventure that is shorter than a full campaign which can vary greatly in length. For the purposes of this answer I will be using the following definitions:

• One shot - an adventure that can be completed in 1 session (generally 4-8 hours)


• Short adventure/campaign - can be completed in 2-5 sessions (4-8 hours per session)


• Long campaign - anything above that

Advantages of a short or one-shot campaign

Testing grounds for character ideas/story ideas/new rules/homebrew

The biggest advantage I have found in my experience running and playing both long and short adventures is that one-shots allow players to play a game and test out character concepts, story concepts, and/or rules variants that are risky or potentially untenable in a long-haul adventure. I've had one-shots turn into longer adventures when the concepts worked out and everyone liked them and also ones where we were happy we tried them out, but were content to stop them after the one-shot was done.

I've run many one-shots where we have tested out risky ideas. The one we are playing now is testing several homebrew rules (one is a neat skill check-based resolution mechanic) and several experimental characters (including me who is testing the new UA alchemist class). Using my character as an example, I never would have wanted to test it out if I knew I was going to be doing a long campaign because it means I would be stuck with a possibly confusing/bad/broken character for a long time or forcing the group to find some way to integrate a new character into the group as I replace it mid-story (we do extensive character work before beginning campaigns so it would be a hassle to say the least).

Great for introducing new players/DMs/systems without a lot commitment

Another really useful way to use one-shots is to introduce new players/DMs to a game. It usually helps, in my experience, to start out new players with a shorter adventure than a full campaign so that they can try an option, learn the rules, and then end the campaign and decide what they want going forward (with regards to character options and the RPG system in general). In particular, I have found this very useful when someone want to try DMing for the first time since they can do so without having the pressure to continue and to reduce the pressure of any mistakes they make.

Every new system I've run or tried with my group has been done as a short campaign or a one shot. We always run one to get a feel for how it plays and if we want to continue playing it. For example, I ran Masks (a Powered by the apocalypse system) as a 2-session intro game before having the group decide if they wanted to continue or not. I gave them a short story to complete but plenty of room to keep going. The group in this case decided to keep going, but easily could have decided that it was enough and to leave it at that. We have two new systems coming up that we are planning to do the same thing with. On the flip side, I've had a player at the table that was new and we started them with a full hardcover adventure in D&D5e. They made a character that didn't really fit, but they didn't understand enough to know that (or express it) and thus they ended up playing 15 levels and two years with a character they would have changed had we run a short adventure first to test things out.

When getting everyone to the table is a DC30 check

One shots are also very handy when you have a very busy group that is only able to meet sporadically. Having an adventure that you can start and finish in one session means that you don't have to worry about the next date people being able to show up at is 2 months after that. It also works in the case that someone has to take a leave of absence and the group wants to break from the current project to wait for them. A one-shot or short adventure can fit perfectly into these gaps.

One-shots are also great for conventions and the like when you are playing with people for only a given time slot and likely never seeing them again.

To give breaks

Sometimes your group is doing a long campaign and the DM just gets burned out for any of many reasons. Sometimes it is even the group that gets burned out as well. Sometimes you just need a short change of tone/pace/system to freshen everything up. One shots deliver this in spades.

A couple examples from my table. Maybe you are in a grimdark 3-year-long campaign and people decide they need a break from it. Maybe the DM gets slammed at work for a month and they need a change so they can get through it but they still want to play. We switched DMs and ran a short 5e one shot in one case and ran a short adventure in another lighter system in the other. In both cases it really helped to have something short and different to switch to without the commitment of a longer campaign. Afterwards, we want back to the long campaign with a fresh take.

Disadvantages of a short or one-shot campaign

It really limits the material and how much you can do

You are limited in the amount of material you can cover in the adventure. Obviously, a short campaign is going to have less stuff in it than a long campaign. But to make a short campaign work it generally has to be very focused such that you can end it in whatever time restriction you set. This often means having a smaller world and a more focused plot with less side quests. The plot may also have significantly less build-up and nuance to it compared with a story that plays out over many sessions.

In my own one-shots I've definitely had issues getting things completed in the alloted time even after thinking that I had made a very focused story. Often I have to railroad the party hard to get them to finish on time and/or cut material. However, this is more an inconvenience that a true bad side and sometimes I have to do this in long campaigns as well for various reasons. It is more pronounced when you are running one shots though for sure, especially if you have a party that enjoys RP/messing around/exploring a lot.

Character development takes a hit

Less character development. Shorter campaign means there is less time to develop party dynamics and characters. Generally, making a character and party takes time. In a long campaign, the players have multiple sessions to feel out their character and discover who they are and what they are like and how they interact with the rest of the party. You don't get as much of that in a one-shot (especially with newer players or players new to a system).

Short testing time

I say that testing ideas is an advantage of one shots, and it is. But, there is an important limitation to it to acknowledge: you only are testing things in a small limited window and that might not generalize into a full campaign context. For example, your UA/homebrew character might work great in your one shot at level 5, but there might be something broken/confusing/unfun that happens at level 11 that completely changes the perspective. One shots give you a taste which is very useful, but it is not replacement for extensive playtesting.

Four solid benefits of a one shot

One shots and campaigns serve different gaming needs, so I slightly disagree with your question's point (in paragraph 3 and at the end) that they are somehow in opposition to each other. They fulfill different needs for the players at the table, to include the DM. A given gaming group can do both!

Four things that a one-shot allows you to do in this game system is:

1. 
   Play at a different level / tier
   
   If you've been playing a Ranger for 4 levels(Tier 1), a one shot can allow you to play (as an example) a 10th level Druid / 1st level Monk (Tier 2) for a few sessions to get a different game experience and see how you like it. I did that a couple of months ago during a play test session. This is a great way to vary and enrich the play experience.


2. 
   Play a different character/class/subclass/archetype to broaden your
   experience
   
   As above, though it need not be in a different tier. We did a one shot a few months back that was a single session. The idea was to help a friend get used to DMing 5e. We all used level 4, Gestalt characters to increase the flexibility of the experience. (And for shennanigans).



3. Give the DM a break
   
   If you are in a campaign, sometimes it is nice to give the DM a break for a session and run a one-shot so that the DM gets to play. I've done this with numerous groups.
   
   This also allows any of the players to try out DM'ing without committing yourself or the player characters to the promise of a full campaign. (Thanks, @Someone_Evil for this excellent point)




4. Test out optional, variant, Unearthed Arcana, or homebrew rules to see if your table likes them
   
   In item two above, we tested out a Gestalt concept within 5e based on the DM's experiences with it in a different edition. Doing that "inside a campaign" would have cost us consistency and continuity.

A potential (minor) down side is: you fall in love with your one shot PC

As you noted in your question, the longer term relationship building and character development is missing from the one shot, and if you fall in love with this PC you have to put it on hold until there is a chance to fold the PC into your campaign, or there is another suitable one shot. But the very act of 'getting into' that character for a session and really enjoying it is fun in and of itself.

Rather than try to do an apples to apples comparison and pick apart the differences, I'll go into some detail about how I've used one-shots (or short-shots) vs campaigns and try to paint them in their best light.

One shot

A one shot is like college: a great time to experiment. Maybe you want to test out some friends that you've never gamed with to see if they mesh with your DM style. Maybe you've got a goofy system that you don't think will stand up to more than a night or two of use. Or maybe the people involved just don't have the time/energy to commit. Don't forget the opportunity to FINALLY dust off that Genasi-Dwarf Valor Bard 6 Old One Warlock 5 Storm Sorcerer 3 you've been trying to play (or whatever).

One shots are not dissimilar to a regular board game night. It's mostly an excuse to get together with some people you like around an activity you're passionate about. The rules of conduct don't have to be as strict as usual (or maybe you can make them even stricter?).

The point is, anything goes; just make sure you clear it with your partners first and everyone will have a great time (usually).

One Shot Example

The last one-shot I ran was with four friends of mine. We've all done board game night together, and I knew one of them used to play D&D (or something similar). my objectives for the game were to vet these people as D&D players (in case I wanted to invite them to a campaign), and also to show them the ropes of one of my favorite hobbies. It was like herding cats. Full murder hobos, and not even the good kind that agree with any plot hook because they know that there will be monsters there; the kind that listens to a mother cry out for the rescue of her child and respond with "nah. eff that. I don't care about any kids. Other character, want to go get drunk in the bar?"

I tried having the NPC interest them with money; no interest. I literally had to say "guys, this is the only plot hook for this. So, if you want to keep playing, find a way to get interested". The barbarian's name was "bork bork". I like these people, and it ended up being a fun night, but I will never invite them to a campaign... well, except that one guy.

Campaign

This is it. You've got the group. You trust them to commit to a few hours every third Saturday. Whether you're the DM or one of the players, you're excited for the group to collectively weave their ways into this wondrous world. /insert epic campaign idea here/

You can look forward to watching a plot unfurl. There could (and should) be twists and turns, emotions can run high. It's like writing a book and watching your favorite show all mixed together. Your character(s) grows, moving from their unsure greenfoot status and finally falling into their own as a mighty champion/devout holy warrior of the great dragon god/cunning master thief that only steals from the rich/whatever.

And depending on your age and inclination, a steady and regular excuse to get together with a few friends who you don't see nearly often enough... it gets harder to do as you get older.

Campaign Example

I currently run one game and play one game. I can expect for each game to play 1-2 times a month, on average. I have a celestial tiefling and there is a very-anti-demon Paladin in the party. We have a Tolkienesque dwarf-elf thing going and it's a fun time. But it started as complete distrust. These characters have grown and learned more about each other.

In the game I run, I can tie in player expectations and drop hints and clues. You have a much broader canvas to paint a picture. I like worldbuilding pretty well, but I like giving players something to look forward to better. The town that the party is in right now: we ended a session outside the gate and I asked two players:

Me (as DM): "what is a rumor you've heard about this town that you think is true?"

Player 1: "the town is inhabited by fish people"

Me (as DM): "ha, okay. and you; what is a rumor you've heard about this town that you think is false?"

Player 2: "the town is inhabited by fish people".

When we regrouped for the next session, I was able to work that into the fabric of the town and we all had a laugh about it, as well as progressing the plot.

Time

As an adult with a full-time job who has many other adult friends with full-time jobs who don't live nearby, one-shots provide a chance to play without a long term commitment. It's nigh impossible to find a weekly or even biweekly time that we could all commit to.

Also, a lot of my friends haven't played D&D before. They wouldn't feel comfortable jumping headfirst into a long campaign. A one-shot gives them a chance to try it out and see if they like it.

New ideas

Apart from that, one-shots give you a chance to do something different that you're curious about but don't think would be fun long term. They also are just a chance to try anything new without a long term commitment.

A break from a campaign

They can also serve as a break from an ongoing campaign. My group had a Halloween themed one shot for Halloween that was a blast.

You could also let someone try DMing or just give the DM a break so they can play for once.