Can anyone explain the exact difference between these? Of course I can visually identify the iota subscript, but I am having trouble understanding why these three kinds of things are distinguished. Is there some kind of rule involved? Does it have to do with the length of the vowels involved? Thanks
A diphtong is two vowels next to each other that are pronounced as a gliding from one vowel to another, e.g. αυ, or an English example, house.
A digraph is two vowels next to each other that are pronounced as one sound, e.g. ου or ει which are sometimes pronounced respectively as a long, closed o and a long closed e. Sometimes they can also be pronounced as diphtongs, which is why these two are called ‘spurious diphtongs’.
The reason why we use iota subscript is that the iota in diphtongs consisting of a long vowel + iota stopped being pronounced sometime in classical times. Sappho or Archilokos would have pronounced them.
Others know much more of this than I and will probably elaborate.
I don’t know if my take on it is anymore accurate than yours but I understood a spurious diphthong to be formed from a contraction or lengthening. (I didn’t realize that it affected its pronunciation though.)
A digraph, in general, is any sequence of two letters used to represent a single sound. “th”, “sh”, “ch”, and “ng” are examples of digraphs in English.
A diphthong isn’t really a digraph, IMHO: although a dipthong is often defined as “two vowels pronounced together as one syllable”, they are still two sounds, not one sound. Two sounds that fit into a single syllable, but still two separate sounds (the vowel plus the “glide” or “semivowel”).
A spurious diphthong, by contrast, IS a digraph. It’s a single sound written with two letters by convention.
(Note that spurious diphthongs were not always written that way; prior to 403 BC, before Athens adopted the Ionian alphabet, spurious diphthongs were spelled simply “E” and “O” – so the verb “to be” was spelled “ENAI”.)
An iota subscript is a convention dating from the Byzantine period. They represent “long dipthongs”, as Anthony said. The convention of “hiding” the second half of the diphthong as a subscript arose from the fact that they were no longer pronounced as diphthongs, but were pronounced simply as the long vowels with no following iota, which is how most people pronounce them now (except for raging, narcissistic pedants, such as myself 
). The ancients wrote in only majuscules (what we call “capital” or “upper-case” letters): miniscules (“small” or “lower-case” letters) were not invented until the Middle Ages. So anything we see now as a spurious diphthong would have been written out in full as a normal diphthong in the classical period. And presumably, were pronounced as diphthongs for much of that time.
If it is a diphtong in ἀκο?ω, how is it pronounced? Like o+u or like o+y (German ü, as I’ve been taught to pronounce it) ? And in case of the latter, isn’t this sound very likely to be confused with οι?
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No, I certainly agree that these changes had taken place by Roman times; in particular, if υ had not changed to ü, The Romans would not have to have borrowed the Greek letter Y (upsilon) to represent it. I was merely surprised that most people on Textkit wanted to push that back to Periclean Athens and even into Homer. Since I couldn’t remember where I’d gotten my impression in the first place, I deferred to the judgement of people who obviously know hundreds of times as much on the subject as I do.