Actually the list of moon-goddesses goes on since, according to Selene's Theoi page:
Other Greek moon goddesses included Pasiphae, the Leukippides, Eileithyia, Hekate, Artemis, Bendis, and Hera (who sometimes doubled for Selene in the Endymion myth).
But let's focus first on the one you mentioned:
- Artemis. Her association with the moon seems relatively straight-forward: if Apollo is associated with the Sun, then his sister has to be associated with the moon.
Selene (= Mene or Luna). She is the proper Titan-goddess of the Moon, often even the personnification of the moon itself. See among many other sources, Apollonius Rhodius's Argonautica:
Rising from the distant east, the Lady Selene (Moon), Titanian goddess...
Phoibe/Phoebe. The Titan Phoibe doesn't actually seem to be associated with the moon, the confusion comes from Selene being often nicknamed Phoebe (i. e. 'bright'). Phoibe was apparently a prophetess Titan-goddess and as such is supposed to have held the Oracle of Delphi at some point. From the first line of Aeschylus's Eumenides:
THE PYTHIAN PRIESTESS
First, in this prayer, of all the gods I name
The prophet-mother Earth; and Themis next,
Second who sat-for so with truth is said-
On this her mother's shrine oracular.
Then by her grace, who unconstrained allowed,
There sat thereon another child of Earth-
Hecate. Now Hecate is something else I think. The different sources we have tend to contradict each other on her parentage, descendance, attributions etc. leading some scholars to think that Hecate was in fact a goddess foreign to the Greek Pantheon that was only added later (and well integrated thanks to the place Hesiod gave her in the Theogony). See for instance what Sarah Johnston says about it in Restless Deads:
Let us go back further, to Hecate's place of origin. There is general agreement that this was Caria, in southwestern Asia Minor, which is supported by the fact that by the Hellenistic period her precinct in Lagina was the largest of all the precincts there (in constrast, in all of Greece, only Aegina seems to have had any significant sanctuary devoted wholly to Hecate).
Caria being in southwestern Anatolia.
Similarly (from the list I mentioned earlier), Pasiphae seems to have been the cretan moon-goddess and Bendis the one from Thrace.
The specific reason why Hecate was associated with the moon seems a bit complex. Originally, in Caria, Hecate seems to have been a fairly generic mother Goddess. Even in Hesiod's Theogony she is presented as having dominion over basically everything:
For to this day, whenever any one of men on earth offers rich sacrifices and prays for favour according to custom, he calls upon Hecate. [...] For as many as were born of Earth and Ocean amongst all these she has her due portion. The son of Cronos did her no wrong nor took anything away of all that was her portion among the former Titan gods: but she holds, as the division was at the first from the beginning, privilege both in earth, and in heaven, and in sea.
According to Sarah Johnston in Hekate Soteira: A Study of Hekate's Role in the Chaldean Oracles and Related Literature (which I haven't been able to find and thus read so I might be misrepresenting her idea here), the association came relatively late (1st Century AD) from the fact that at some point Hecate was seen as a "liminal" goddess, an intermediary between two worlds (hence also probably the association between Hecate and witchcraft), which is a quality that was supposedly shared by the Moon.
So to summarize, the difference between Selene and Artemis is that one is the moon while the other is just associated with it, and on top of that one is a Titan while the other is an Olympian; Phoebe has nothing to do with the moon; and the others are foreign/regional goddesses integrated a posteriori into the Greek Pantheon.