Although there do not appear to be any ancient sources which specifically address this question, the most likely reason for the disappearance of female praenomen seems to be the absence of a role for women in public life (as Mark C. Wallace noted in his comment above), a small number of aristocratic ladies excepted.
Also, the increase in the use of distinctive cognomina for women helped to render praenomen unnecessary. It's also worth noting that, while male praenomen did not disappear, they were increasingly abbreviated to a single letter, their importance confined to when a distinction between males in public life was needed. Within the family, though, they would have been of little use when sons bore the same praenomen, as was the case with the emperor Vespasian (Titus Flavius Vespasianus) and his elder brother Titus Flavius Sabinus.
As to why Greek women had them but Roman women didn't, this was probably due to social and /or cultural and / or regional influences, but the picture is very complicated (as can be seen below). Also, it appears some Roman freedwomen did have praenomen, as did some Roman freeborn women in Greece who Hellenized their names.
While I have been unable to access what appears to be the main academic source on female naming - Roman Female Praenomina. Studies in the Nomenclature of Roman Women by Mika Kajava (1994) - I have found two reviews (both in French) of this tome. Mireille Cébeillac-Gervasoni, in her review in Latomus, T. 59, Fasc. 1 (2000), writes that Kajava
...cherche à résoudre un problème déjà débattu et resté sans
solution : l'absence de prénom de la femme alors que les hommes en
portaient un et elle suppose que l'explication logique pour cette
carence est qu'on jugeait - sauf pour quelques cas de grandes dames de
l'aristocratie - que ce prénom était inutile pour distinguer des
femmes sans rôle dans la vie publique.
Translation:...seeks to solve a problem that has already been debated and has remained unsolved: the absence of women's first name while men had one, and she assumes that the logical explanation for this deficiency is that it was thought - except for a few cases of great ladies of the aristocracy - that this first name served no purpose for distinguishing women as they had no role in public life
Monique Dondin-Payre's review of the same book essentially says the same, but with a little more detail which highlights the complex and (for us at least) confusing use of names in Roman times:
Contrairement aux prénoms masculins, ils sont strictement réservés à
la sphère privée (sans qu'il y ait eu interdiction d'emploi public)
puisque les femmes n'ont aucun rôle civique; c'est pourquoi ils sont
rares, non officiels, peuvent disparaître ou apparaître pour une même
personne au gré des besoins
Translation: Unlike male first names, female ones were strictly reserved for the private sphere (without there being ban on their public use) since women had no civic role; it is why they were rare, unofficial, and may disappear or appear for the same person according to needs
Benet Salway, in What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700 (The Journal of Roman Studies Vol. 84 (1994), pp. 124-145) has this to say:
Since women normally retained the paternal gentilicium, outside the
family context these alone frequently sufficed to distinguish them; if
not, their father's or husband's name in the genitive might be
appended. Hence the early demise of the feminine praenomen. Within the
family a system of naming successive daughters Secunda, Tertia,
Quarta, etc. or Maior and Minor appears to have arisen.
However, this trend did not continue:
Towards the end of the Republic and moving into the Principate,
daughters were less and less being given impersonal numerals to
distinguish them from their sisters. Now they might be distinguished
from their sisters by having received cognomina from either paternal
or maternal sides of their family, present or past generations, often
in the feminine diminutive and formed with the suffix "-ina" or
Other possibilities for distinctive feminine cognomina included the
addition of complimentary names such as "Pulchra" ("beautiful"), or
romantic names of Greek origin...
Thus, other naming practices rendered the praenomen redundant in private life (and doubtless nicknames were also used, as T.E.D. noted in a comment above), while in public life they were usually only needed by men (Livias and Agrippinas excepted!).
Why Greek women (non-Romanized - the relevance of this distinction will become evident shortly) had personal names while Roman women didn't is not dealt with directly by any of the sources I've found, but there are nonetheless a few points worth making on this. The first is that the Roman praenomen was always scarce:
...in stark contrast to Greek practice the Romans used a very small
selection of personal names and the weight of convention seems to have
strongly discouraged indulging in new coinages. Ninety-nine per cent
of Romans of the regal and republican period shared one of only
seventeen praenomina. Women were deprived even of this limited
While this doesn't directly address the issue specifically in relation to different female naming practices among Greeks and Romans, the fact is that praenomen became unimportant for both Roman women and men; true, women lost them more than men (though not completely - see below), but for men they were usually abbreviated to a single letter and were of marginal importance except in specific circumstances. As the praenomen was 'relegated' to a position of relative insignificance for most Romans, I think it's fair to argue that the difference between Greek and Roman women in this regard is also relatively insignificant, and this argument is perhaps supported by the observations below. What follows comes primarily from Female Tria Nomina and Social Standing in Late Republican and Early Imperial Periods by U. Kantola & T. Nuorluoto (2016)
Interestingly, there were differences in female naming practices between freeborn women and freedwomen in Italy; the latter often had praenomen whereas the former usually didn't. There is thus the possibility that not having a praenomen was 'used' by some elite families as a means of setting them apart from 'lower' social classes.
Also of interest is that freeborn Roman women in Italy and those (Roman and Romanized) in Greece had different naming practices; the latter were more far more likely to use praenomen than the former as some Roman women in Greece 'Hellenized' their names. This would seem to contradict the argument in the previous paragraph (i.e. why would Roman women 'lower' their social status by adopting praenomen when these were mostly used by freedwomen?), unless one considers that there existed different conventions in different environments, and that Roman families in Greece were perhaps more concerned with 'blending in'. Also, we cannot assume that there was a consensus among Roman families (even just with Rome) as to the 'social status' of the female praenomen.
In summary, it's very complicated (again!) and we also have to be aware that the mostly epigraphic evidence presented in the previous two paragraphs is based on a fairly small sample of names.