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Compare the following two Google Ngram Viewer charts for sceptical vs. skeptical in American English and British English:

British English

American English

My interpretation of these charts is that:

1. Before the 1910-1920's, sceptical was used in both American English and British English more than skeptical.




2. Something happened in the 1910's in American English that made the usage of skeptical become much more common than sceptical. This change never happened in British English.

Why did this change happen in the US? What was its trigger?

Note that while etymoline.com states that the sk- spelling is preferred in the US, it does not explain why it became so more common than the sc- spelling, in particular since the 1910-1920's.

skeptic also sceptic, (...)
The sk- spelling is an early
17c. Greek revival and is preferred in U.S.

The reason? A man by the name of Noah Webster, who wrote America's blue-backed spellers, and her first dictionary.

Noah Webster, was an English spelling reformer, and one of the chief advocates of English spelling reformers is that spelling should change alongside pronunciations :

Pronunciations change gradually over time and the alphabetic principle that lies behind English (and every other alphabetically written language) gradually becomes corrupted. If the maintenance of regularity in the orthography of English is desired, then spelling needs to be amended to account for the changes.

This change was made along with many different words (e.g. colour to color, grey to gray, -ise to -ize)

"Sceptical" changed to "skeptical" due to Noah Webster's spelling reforming efforts, basically.

Following Thursagen's trail on Noah Webster, I found the following note in 1828 Webster's entry for skeptic

This word and its derivatives are often written with c instead of k in the first syllable, -- sceptic, sceptical, scepticism, etc. Dr. Johnson, struck with the extraordinary irregularity of giving c its hard sound before e, altered the spelling, and his example has been followed by most of the lexicographers who have succeeded him; yet the prevalent practice among English writers and printers is in favor of the other mode. In the United States this practice is reversed, a large and increasing majority of educated persons preferring the orthography which is most in accordance with etymology and analogy.

Early 'scepticks' and 'skepticks'

The earliest occurrence of sceptical (or skeptical) that I've been able to find in an English dictionary is in Edward Phillips, The New World of English Words, or a Generall Dictionary (1658):

Sceptical, (Greek) contemplative, whence Scepticks are a sort of Philosophers who onely consider and contemplate of things without determining any thing.

John Kersey, The New World of Words: or, Universal English Dictionary (1706) has this series of relevant entries:

Sceptical or Sceptick, belonging to the Scepticks or Scepticism ; that is in doubt or suspence, doubtful.

Scepticism, the Doctrine or Opinion of the Scepticks.

Scepticks, a Sect of Philosophers, who contemplated and consider'd Matters, but doubted of every thing, and would admit of no Determination : Whence the Term is apply'd to a Person who maintains there is nothing certain, and no real Knowledge at all to be had.

To like effect is Kersey's successor, Nathan Bailey, An Universal Etymological English Dictionary, second edition (1724) [you have to scroll down to the sceptical entry, unfortunately, because Google Books refuses to find matches from the relevant page]:

SCEPTICAL, SCEPTICK Sceptique, F. Scepticus, L. Σχεπτχις, Gr. belonging to the Scepticks or Scepticism, Contemplative, that is in Doubt or Suspence, Doubtful.

SCEPTICISM, the Doctrine and Opinions of Scepticks.

A SCEPTICK, Sceptique, F. Scepticus, L. of Σχεπτχις, Gr. of το Σχεπτεσϑ Gr. to look out or observe, to contemplate a Sect of Philosophers who contemplated and considered Matter, but doubted of every Thing, and would admit of no determination, thence the Term is applied to those who maintain there is nothing certain and no real Knowledge at all to be had, but that a Man ought to doubt and disbelieve every thing.

Oddly enough, the first dictionary to use the spelling use of skeptical is Nathan Bailey, An Universal Etymological English Dictionary, "The Second Edition with many Additions" (1731), which provides full entries for scepticalness, scepticism, and scepticks, but also full entries for skeptic/skeptick, skeptically, and skepticism, the last defined as follows:

SKEPTICISM, the doctrine and opinions of the skepticks ; which was, that Persons ought to suspend their judgment, as to the determination or firm belief of any thing.

Samuel Johnson, who seems to have been no admirer of the ancient skeptics and who famously attempted to refute the more recent Bishop Berkeley school of doubt by kicking a stone, weighed in on the side of skeptical. Here are the relevant entries in Samuel Johnson, A Dictionary of the English Language, volume 2 (1756):

SCEPTICK. s. See SKEPTICK.

...

SKEPTICK. s. ςκεπλομαι One who doubts, or pretends to doubt of every thing. Decoy of Piety. Blackmore.

SKEPTICAL. a. from skeptick. Doubtful ; pretending to universal doubt. Bentley.

SKEPTICISM. s. Universal doubt ; pretence or profession of universal doubt. Dryden.

Johnson's preference was far from universal, however, as is evident in the latest effort from the ever-unpredictable Nathan Bailey, The New Universal Etymological English Dictionary, fourth edition (1756), which dumps all of the entries with k as the second letter in favor of entries for sceptical, scepticalness, and scepticism.

'Sceptics' and 'skeptics' in the New World

I was somewhat surprised to find that Noah Webster, A Compendious Dictionary of the English Language (1806) ignores the k spelling altogether:

Sceptic, n. one who doubts of every thing, especially of the truth of revelation

Sceptical, a. doubting, hesitating to admit the truth of an opinion or system

Sceptically, ad. in a sceptical manner

Scepticism, n. doubt, hesitation to admit the truth of revealed religion

Scepticize, v. t. to doubt, to be sceptical.

Likewise Webster's first full-size An American Dictionary of the English Language (1828) provides entries for the same five spellings as the Compendious Dictionary did, plus a brief cross reference:

SKEPTIC. See Sceptic.

Webster announces his break with the form sceptic in his July 1841 preface to An American Dictionary of the English Language, second edition (1846):

In a class of words which have borne two forms, the author selects that which he deems most proper, and discards the other. Thus he prefers to write afterward, backward, forward, onward, toward, &c, without the s. He rejects amongst and whilst, as obsolete ; and disannul, as an unauthorized and unnecessary substitute for annul. He prefers skeptic to sceptic ; gimlet to gimblet ; Mohammedism to Mohammedanism ; chamomile to camomile ; handcraft to handicraft ; handwork to handywork ; incase to encase ; enlist to inlist ; embody to imbody. ... On the principles laid down in the Preface to this Abridgment, most of these words were inserted under both their forms ; and still are suffered to stand because it was found difficult to make the change. It will be understood, however, from this statement, which form the author approves.

The dictionary itself retains the cross reference "SKEPTIC, See SCEPTIC" under the alphabetic entry for skeptic. But rather bizarrely when you turn to the place where you'd expect sceptic to appear—namely, between the entries for scentless and sceptre—you instead find full entries for skeptic, skeptical, skeptically, skepticalness, skepticism, and skepticize.

Finally, the "New Revised Edition" of 1847—the edition of the dictionary issued following the death of Webster and the acquisition of the rights to his dictionary by the G. & C. Merriam Company, moved the skeptic entries to their correct alphabetical location and signaled the demise of sceptic with the entry there of the simple cross reference

SCEPTIC. See SKEPTIC.

Conclusion

Whatever train of thought led Webster to abandon sceptic in favor of skeptic, it was by no means an original idea, as skeptic had begun appearing in English dictionaries for more than a century before Webster embraced it. Nevertheless, it seems extremely likely that the dominance of Merriam-Webster's dictionaries in the American market from 1850 forward contributed greatly to the ascendency of that spelling in the United States.

The letter group sc can be read /s/ as in scene and /sk/ as in sceptical.
So the American spelling is much clearer. The British spelling is as in French and Latin, the American spelling is as in the original Greek word skeptikos.

Actually, I think it was the break between English (British side) preferring to keep the Francophone spellings of words borrowed from France, and America, seeking to reduce the special rules needed to remember how to spell words having distinct pronunciation agreement.

American 'skeptic' <-- British 'sceptic' <-- French 'sceptique' <-- Greek 'σκεπτικιστής' ... because the French simply didn't like K's.

The situation is similar for the '-our' words Behaviour, behavior; Colour, color; but oddly not for Manor, manor. By dropping the French spelling affectations, American English became easier to learn and use correctly. On this, I applaud ol' Noah Webster.