In the orthographie of modern French, the little hat (^) sitting atop an e suggests that, in days of yore, when French sounded a lot more like Spanish than it does today, the vowel in question preceded an s. Thus, among other things, the accent circonflexe over the e in fête reminds us of the festive root of that word.
In England, where fête has come to mean an outdoor party of the family-friendly sort, the first e in the word retains its headgear. (‘Digby won a bottle of home-made elderberry cordiale tossing bean bags at the village fête.’) In America, however, the word will often appear sans châpeau. (Eliot House, at Harvard, which hosts a spring formal called Fête, provides us with the exception that proves this rule.)
When they use it as a noun, les Anglo-Saxons are doomed to pronounce ‘fête’ in a way that resembles a word meaning ‘wyrd’. When, however, we use ‘fête’ as a verb, we often say it in a more Gallic manner. Thus, when ‘we fête Digby for the virtuosity he displayed tossing bean bags at the village fête’, we pronounce the second instance of the word as ‘fate’, but the first as ‘fett’.
For Further Reading:
A fete accompli.