So, here’s the mainstream answer. 🙂
Greek had iota, eta, and upsilon as different letters, because they used to be pronounced differently:
- Iota was always an /i/
- Eta was a long /ɛː/. In fact, in many archaic variants of the Greek alphabet, it was written as an epsilon /e/; that was the case in Athens before 404 BC, when they adopted the Ionic variant of the alphabet, which had the eta as an /ɛː/ instead of a /h/. Eta switched from /ɛː/ to /i/ some time around 300 AD, although there are dialects of Greek where an /e/ value survives for eta (notably Pontic).
- Upsilon was probably originally /u/; in Attic it was /y/. The switch to /i/ is very recent: we have a poem from 1030 AD where a priest is made fun of for pronouncing ξύλον as ξίλον. There are dialects in which a /y/ pronounciation survives as /ju/; they include Tsakonian, Old Athenian, and Maniot.
That’s the history. Modern Greek uses those letters because its orthography is historical: if a word was spelled in Ancient Greek with an eta or an upsilon, it still will be spelled with one in Modern Greek. A lot of the controversy around modern Greek orthography in the early 20th century was because philologists had to work out the etymology of modern words, to work out how they should be spelled. Hence the switch of “to watch” from κυττάω to κοιτάω.
There was also a tendency to use the old variant i’s anachronistically, in transliterating loanwords and foreign names. A long /e/ would be transliterated as αι; a long /i/ as η; a long /o/ as ω; a spelling <y> as υ. So Σαίξπηρ <Saixpēr> for Shakespeare, which is just pronounced /sekspir/. That tendency has been abandoned the last few decades. I spell train as τραίνο <traino>, but I’m old; the spelling being taught now is τρένο <treno>. The ratio of the old-fashioned Χίλαρυ <Chilary> to the phonetic Χίλαρι <Chilari> online for Hillary is 1:50.