"Oestrogen" and "oesophagus" -- why are they spelled differently in British English?

Solution 1:

The word oestrogen comes from the Latin word oestrus, and oesophagus is a Latin word as well.

The oe spelling in Latin originally represented a diphthong [oj] ("oy"), but then later (in Latin) became a long vowel [ee]. When we borrowed such words into English, it was pronounced more like [i], [e], or [ɛ] (depending on the word), following English pronunciation rules.

And since we pronounce them like "estrogen" and "esophagus", the US English spelling was changed to reflect the English pronunciation more closely, by dropping the o. In British English, the connection to the original Latin spelling was retained.

(Something similar to this happened with, e.g., encyclopaedia (British) / encyclopedia (US).)

Solution 2:

The person most credited with these systematic changes in American spelling is Noah Webster. He developed and published the first complete American English dictionary about 200 years ago. He was a believer in spelling according to sound and consequently removed a number of silent letters. He also changed "s" to "z" in certain words (realise" to "realize" e.g.) And finally, he changed the "-re" ending found in French-derived nouns to "-er". ("theatre"-"theater")