In my textbook for the perfect tense 3rd person plural, I have learned forms like habuērunt and amāvērunt but found a conjugation table where there are also slightly different forms like habuēre and amāvēre. Are these alternate forms completely interchangeable or are the two forms used in different grammar patterns? Do they represent different dialects or forms specific to different periods of history? Just curious 
Functionally it is simply an alternative ending. It’s archaic, but used particularly in poetry (I seem to recall that Vergil in the Aeneid has the ending over 200 times) perhaps to give it that fine old fashioned feel, but also because it allows a bit more flexibility in meter. I recall seeing it in Livy (don’t ask me where!) and a few other prose authors, I suspect for rhetorical effect.
And, an interesting article on archaisms in Vergil’ Aeneid which touches on the use of the alternative form:
https://projects.iq.harvard.edu/persephone/aeneid-and-archaism-0#_edn20
Particularly:
Nevertheless, it remains true that Virgil was amantissimus vetustatis, for otherwise it is unconceivable, for example, why the poet would wish to use the 3rd person perfect plural ending -ēre, which is archaic and has a ritualistic tone, far more frequently than the endings -ěrunt/-ērunt.[xx] Here, one can argue that Virgil does accommodate some old morphological forms into his new Dichtersprache, and when he does so, he respects them by using them frequently. This may have a thematic correspondence to Aeneas’ accommodation of the past into the present, which one can discern, for example, in the imagery of the colours gold and purple and its Oriental, ‘Eastern’ association: Aeneas uses one of the two robes of gold and purple made by Dido (cf. 11.72-75) to wrap Pallas’ body and keeps the other for his personal use,[xxi] symbolically depicting the suppression of his Oriental identity [his Trojan origin, i.e. the past] but also its continuation in the Roman background [i.e. the present]. Virgil’s use of archaism may suggest that, just as the language of Republican poetry before the development of the neoteric school must continue to be respected but in a suppressed manner, likewise the image of old Rome and her relationship with old Italy must not be erased entirely as the Romans move onto their next chapter of politics. Therefore the first question I posed – whether Virgil’s use of archaism has any thematic significance in terms of temporal relationship – seems more than possible, although to prove this point is impossible: one would need to know how Virgilian Latin was received by his early readers, but there is simply not enough evidence on this matter. Moreover, and crucially, our evidence for Roman literature, especially pre-Virgilian literature, is too fragmentary to derive any secure linguistic argument.
Hi, just to add, the two endings do not come from different dialects: they have a complex but single history. The place to start if you’re interested is sec. 429 of Sihler’s New comparative grammar of Greek and Latin. Sihler notes there that ‘there is more complexity in the history of the 3pl. than in all the rest of the endings together’, and that in the parent language of Latin:
the 3pl. forms in *-> nt > correlate with 3sg. forms with *-> t> , whereas the > t> -less 3pl. forms (that is, the ones in → r> ) correlate with > t> -less 3sg. In the stative, all 3rd person forms were uniformly > t> -less, in contrast to the jumble of competing inflections in the eventives; this may account for the clearest survival of the > r> -forms in the stative paradigm, though even there it was subject to encroachment by the > nt> -types and their reflexes …
The two endings may ultimately trace back to one and the same ending deeper in history (sec. 429(b)), but after they split into different endings in the parent language, the
→ nt > endings generally spread at the expense of the > r> -endings, ousting the latter completely in G
(sec. 429) (where “G” stands for Greek).
Sihler is heavy reading, but can be very helpful to see the patterns underlying the endings and stems in both Greek and Latin.
Cheers, Chad
And just for fun, we see two nice examples in Ovid 4.55ff,
“Pyramus et Thisbe, iuvenum pulcherrimus alter,
altera, quas oriens habuit, praelata puellis,
contiguas tenuere domos, ubi dicitur altam
coctilibus muris cinxisse Semiramis urbem.
Notitiam primosque gradus vicinia fecit:
[60] tempore crevit amor. Taedae quoque iure coissent:
sed vetuere patres.
Grātiās tibi agō! Laetus sum