From the 2nd century BC to the 20th century AD, Greek spelling used three different accent marks, but there has always been only one phonological accent, which nowadays surfaces as an intensity-based stress, but was undoubtedly pitch-based in ancient Greek, even if some details aren’t completely clear. The location of that accent has been stable for millennia. Even such details as doubly-accented proparoxytona before enclytics have been preserved: modern Greek has (in the single-accent orthography adopted in 1982) η πρόσκλησή μου (my invitation), with stresses on the same vowels as in ancient Greek ἡ πρόσκλησίς μου (my subpoena: the meaning did change... ) The exceptions (e.g. ἐλευθερία becoming λευτεριά) need not bother us here.
It is a trivial fact that, in probably all languages of the world past and present, words in connected speech are pronounced differently than their citation forms. So, the final mora of ancient Greek oxytones was realised before other accented words with a lower pitch, probably but not surely the same low pitch of a non-accented mora. However you call it (accent suppression might be a good term), it was an entirely predictable phenomenon, and an integral part of the realisation rules of the ancient Greek pitch accent. An abstract phonological representation of the system might even ignore the difference. My first answer is therefore “There was no phonological difference, the point is moot”.
However, the ancient grammarians were no modern linguists and had a musical ear: they decided that what was happening was noteworthy enough to use a special accent for it, the grave.
And there is more. In modern Greek, you cannot have two consecutive stressed syllables. So, even if the modern orthography obscures this fact, e.g. the last vowel of γιατί (why) is fully stressed in δεν ξέρω γιατί (I don’t know why) and practically unstressed in δεν ξέρω γιατί κάνεις αυτά (I don’t know why you’re doing this). My second answer is therefore “The change from pitch to stress changed the conditions, but accent suppression has never gone away and is still there”.
In their article from 1985 “Stress in Greek?”, A.M. Devine and Laurence D. Stephens debunk Sidney Allen’s theory that ancient Greek had a “non-accentual stress” in addition to the pitch accent talked about so far. Then, they try to determine how and when the modern stress accent developed. They analyze spelling errors through the centuries and conclude that
These results lend some support to the view that a stress accent developed before the loss of quantity contrasts and conditioned that loss89.
89 False quantities begin to appear with some frequency in Christian and popular verse during the third century A.D. At this date they may not be motivated by contextually limited phonological factors such as accentuation, but merely reflect imperfect mastery of an artificial literary register by speakers who have lost all traces of quantity distinctions in all environments.
I would also like to cite another, more recent article from 2011. I don’t know how well it was received and if it is still controversial, but I found it fascinating:
Mindaugas Strockis (Vilnius University), A New Interpretation of the Syllable Tones in Doric Greek. It is well known that in many cases the moras marked as accented in Doric occurred after the corresponding Attic ones: compare Doric φρᾱτήρ with Attic φράτηρ, Doric Ποτιδάν with Attic Ποσειδῶν, Doric σκώρ with Attic σκῶρ, etc.
According to Strockis, this doesn’t mean that the Doric phonological accent was displaced with respect to Attic: it was at the same place, but it was realised as a pitch fall instead of a pitch rise: the acutes and circumflexes would then mark a pitch rise after that fall, effectively marking a post-tonic mora. If he is right, the purely musical meaning of the ancient notation would be confirmed, and our understanding of the whole matter would be considerably deeper. I add a fact not mentioned by Strockis, but perfectly in line with his theory: Tsakonic, the modern dialect derived directly from Doric, has its stress accents exactly where they are in the dialects derived from Attic.
Finally, I hope that it is not too off-topic to mention that in recent times the grave was considered nothing more than a typographical habit, of the kind of σ vs. ς: it was the first mark to go away, well before 1982. You can find a lot of 20th-century publications with circumflexes, spirits and all, but no graves: acutes are used instead. Some traditionalists use this “simplified polytonic” system to this day.