Do Americans also typically use the word "aesthetic" spelled that way?

As far as I know, the word "aesthetic" can be considered the "British" or "European" way of spelling the word, like "caesium" or "haemophilia". The spelling "esthetic" (which replaces the ae with e as so many American spellings of those words do) does exist, but I've barely ever seen it written that way anywhere. Do Americans also typically use "aesthetic"? If so, how come?

It is true as a general rule, that with words that contain an ae or oe originating from a dipthong of Latin or Greek origin (rather than from a compound of a word ending in a or o and one beginning with e, or from the letter æ as used in some Germanic languages), then:

1. You will most likely find the spelling to be ae or oe in some forms of English, including British, and Irish English.
2. You will most likely find the spelling to be e in some other forms of English, including American and (to a lesser extent) Canadian.
3. You will most likely not find the spelling to be æ or œ any later than the early twentieth century, though it does still occur in some idiolectic orthographies (hey, some of us just think it reads nicer).

Famously, moving toward the simpler e was one of Webster's spelling reforms, which as a rule were more popular in the US than elsewhere (though neither fully accepted there, nor fully rejected elsewhere), but those forms were found before his time, and he settled on them as the preference rather than inventing them.

There are though a few exceptions.

In one direction you've noticed that aesthetic is more common than esthetic as a spelling of æsthetic, even in the US.

Just why is hard to say, though probably a factor is that moving toward esthetic would change the beginning of the word, and so be a strong change in people's minds (and so less likely to seem "natural") and affect where it was found in the dictionary. Likewise, oenology is more common than enology in the Americas.

Conversely, fœderal and foederal are pretty much dead, being replaced by federal in British as well as American English. Likely this owes something to fact that the term federal is of such import in American politics and fœderal has not seen much currency in British politics since the Treaty of Union of 1706 and Acts of Union 1707, during which time the term fœderal union was a popular description of the proposed new Kingdom of Great Britain: When those of us in Britain and Ireland hear about "federal government" it is most often in the context of American matters. (Australians also have a federal rather than foederal government, foederal had already died out by the foundation of the Federation of Australia).

In the middle, fetus will be found alongside foetus and fœtus in Britain (foetus and fœtal are a hypercorrection anyway, as the original Latin was fetus though the other forms started popping up in Latin texts around the 16th Century), and likewise homeopathy for homoeopathy or homœopathy, and medieval for mediaeval or mediæval can be found in Britain. Conversely archaeology is found in the US outnumbering archeology.

The reasons are various and often hard to trace. In some cases a standard may have an effect cesium is most common in the Americas, but some scientiest use caesium as the IUPAC-recommended spelling. In some cases it's a matter of who one sees writing of a particular topic the most; there is after all commerce of ideas between the English-speaking countries. In many cases, it's really hard to say just why a form became preferred where it bucks the general trend.

Classical names are generally in the longer form, such as Caesar, while current names in the form used by their personal or corporate owner; hence Encyclopædia Britannica, World Book Encyclopedia and Chambers's Encyclopaedia so as to remain faithful to what is seen as the proper form of the word in each case, under the general rule that one should avoid orthographic damage to someone's name.

Ngram evidence

Let's start by looking at U.S. preferences in the spelling of the aesthetic/esthetic/æsthetic family of words over the years. Here, to begin, is an Ngram chart for aesthetic (blue line) versus esthetic (red line) versus æsthetic (green line) for the corpus American English over the period 1800–2005:

Here is an Ngram chart for aesthetics (blue line) versus esthetics (red line) versus æsthetics (green line) for the same corpus and time period:

And here is an Ngram chart for aesthete (blue line) versus esthete (red line) versus æsthete (no line, due to insufficient instances of the word) for the same corpus and time period:

Several things are evident from these charts. First, and most obviously, the æ/ae spellings have always (except for a one-year blip in 1847 in favor of esthetic, and a spurious uptick in favor of esthete in 1804 based entirely on OCR errors) been more common in U.S. usage than the e alternatives. Second, the frequency lines for aesthetic and aesthetics rose fairly steadily from 1850 or so until the 1990s; and the line for aesthete, though it plateaued about mid-century during the 1900s, has held its own since then. In contrast, the e forms, after rising in frequency between 1900 and 1940 or so, flattened out and then began to decline substantially during the past half-century. Third, none of the words in this family show consistent year-to-year use until the 1830s at the earliest. And fourth, Google Books can't be relied on to distinguish between æ spellings and ae spellings

This last point is noteworthy because it indicates that aesthetic/esthetic was not an established word in U.S. English during the earliest decades of the 1800s—and this is significant because that was the period when æ spellings were at greatest risk of being altered by Noah Webster in his pursuit of simplified orthography.

The historical trouble with 'æ'

Samuel Johnson was hardly warm toward æ in his original Dictionary of the English Language (1756), as his entry for the diphthong/character indicates:

AE, or Æ. A diphthong of the Latin language, which seems not properly to have any place in the English.

Despite that unwelcoming description—and despite avoiding æ in his spelling of such entries as archailogy, cesarian, etiology, and hemorrhage—Johnson includes entries for (among other terms) ægilops (a tumor of the eye), ægyptiacum (an ointment), æthiops (a mineral), and ætites (another mineral). Johnson also includes various entries in italic type (to indicate their enduring foreignness) such as cæcias ("a wind from the north") and cæsura (a figure in poetry").

Noah Webster, An American Dictionary of the English Language (1928) seems to take the denunciation a step farther in that dictionary's entry for æ:

Æ, a diphthong in the Latin language ; used also by the Saxon writers. It answers to the Gr[eek] αι. The Sax. æ has been changed into e or ea. In derivatives from the learned languages, it is mostly superseded by e, and convenience seems to require it to be wholly rejected in anglicized words. For such words as may be found with this initial combination, the reader will therefore search under the letter E.

Nevertheless, Webster introduces two terms starting with æ that do not appear in Johnson's 1756 dictionary: ædile and ægis. He also retains ægilops and cæcias, though he changes æthiops to ethiops, ætites to etite, and cæsura to cesura/cesure (ægyptiacum vanishes altogether). Given the inconsistency of Webster's handling of æ words, it's difficult to say whether he would have given æsthetics/esthetics the ægis treatment or the etiology treatment if he had chosen to include the word in his 1828 dictionary.

The ascendancy of 'æ/ae'

When the word did finally appear in a Webster's dictionary, it was in the Merriam-Webster edition of 1847 (shortly after Noah Webster's death), and that dictionary's opposition to æ words was noticeably weaker than, for example, the 1828 Wenster's was: the 1847 American Dictionary of the English Language reinstated ætites while dropping etite, added entries for ætiology and cæsura (with definitions) while retaining separate entries (with definitions) for etiology and cesura, and provided new entries for æsthetics/esthetics, esthetic, and esthetics, as follows:

ÆSTHETICS, ESTHETICS, n. The theory or philosophy of taste ; the science of the beautiful, or that which treats of the principles of the belles lettres and fine arts.

...

ESTHETIC, a. Pertaining to the science of taste.

ESTHETICS, n. The science which treats of the beautiful, or of the theory of taste. {See ÆSTHETICS.}

The 1847 Webster's willingness to countenance both æsthetics and esthetics—and indeed to put the more detailed definition under the entry for æsthetics/esthetics—while insisting that the adjective form must be spelled esthetic doesn't make a lot of sense historically. After all, the earliest match for the word in Google Books search results is from a 1797 translation of selections from the philosophy of Emmanuel Kant as expounded by James Beck, which refers, three times, to the "transcendental aesthetic" (as a thing fundamentally distinct from the "transcendental logic"). Most of the Old World occurrences of the word æsthetic/aesthetic/esthetic that found their way to the United States in the first half of the 1800s appear to have used either the æ or the ae spelling, although instances of esthetic did sometimes occur, as in this example from Flower's Political Review and Monthly Register (January 1812):

Kant has founded a new esthetic school, in his " Criticism on the Judgement."

One of the earliest home-grown instances of the word in U.S. English is from William Woodbridge, "On the Education of the Eye" in The American Monthly Magazine (July 1838), which uses the spelling æsthetic (twice).

We must avoid the extremes into which Herbert fell, in confounding the moral sense with taste—in attempting to make beauty the basis of virtue. The one is occupied with moral relations, the other with sensible objects. Still the æsthetic principle, when fully developed, is a material aid to the moral one. The connexion of these subjects is like many other facts in our nature, not easy to explain, but not the less certain. Men of refined taste do not fall into gross vice, at least, so easily as others. ...

On this subject Fellenburg observes, "For those who despise exercises in the arts, I could only wish the incontestable truth were impressed upon their minds, that a well-formed taste, a delicate æsthetic judgment, although it can never supply the defect of religious feeling, affords, on many occasions in life, more assistance to human weakness than the colder convictions of moral duty; ...

This extract combines æsthetic with other spellings that—inconsistently—are either primarily British (connexion) or primarily American (judgment). The preference for æsthetic or aesthetic only increased in the second half of the nineteenth century. By the time Webster's [First] International Dictionary appeared (in 1890), the e spellings appeared as a joint entry that served as a cross reference pointing to the æ spellings for the actual definitions.

The Simplified Spelling Board and 'esthetics'

In the early 1900s, a reinvigorated movement to alter the standard spellings of many English words caught on in the United States to the extent that President Theodore Roosevelt seems to have instructed the U.S. Government Printing Office to adopt "rational spelling" (as defined by the Simplified Spelling Board). An article by Lee Henry, "After Thoughts on Spelling Reform," in Common Sense (March 1907) conveniently lists the lists the 300 simplified spellings endorsed by Roosevelt, which include esthetic and esthetics. (A lengthier discussion of the changes—and ad-man-on-the-street reaction to them—appears in "'Simplified Spelling'—The Day of Abbreviation," in Men's Wear (September 5, 1906). As Harper's Weekly (September 8, 1906) notes, many of the simplified spellings were already the more common forms in U.S. English—but that was clearly not the case with esthetic and esthetics.

The improved fortunes of esthetic, esthetics, and esthete in the Google Ngram charts shown above track the fortunes of the simplified spelling movement, which succeeded with ax, clue, ether, fantasy, licorice, sulfur, and woolen, but failed rather miserably with altho, fantom, gazel, profest, thoroly, thruout, vext, and wo. The e versions of the esthetic group achieved middling success until about 1970 when, the simplified spelling movement having lost most of its steam, their fortunes declined significantly.

Meanwhile, the rise of the typewriter keyboard caused the æ versions of the same words to lose considerable ground to the alternative ae spellings, because the limited number of keys available pushed manufacturers to give up a dedicated Æ/æ key in favor of other needed special characters, forcing typists to replace those characters with the two-letter Ae and ae equivalents. And that's where we appear to be today.

Conclusions

The notion that esthetic/esthetics/esthete were ever the preferred U.S. spellings of this trio of words simply isn't supported by the available evidence. They arrived too late into common U.S. English to be subjected to Noah Webster's seemingly haphazard efforts to suppress æ spellings in favor of e alternatives; and the simplified spelling movement, though it did reinvigorate the e spellings, never pushed them into equal popularity with the competing æ/ae variants in the United States. And as the results for the period 1960–2005 on the three Ngram charts above indicate, esthetic/esthetics/esthete are far less popular than their ae alternatives today.