As i have understood it, it depends on the forms of cults or the setting/purpose in which the god was worshipped, but I would still here more about it. Were there regional differences and what were they? Were there "competing" cults?
The Epithets sometimes have nothing to do with what the God is actually doing or representing at the moment, the classic is how Achilles the Fleet Footed is lying in bed wailing. Epithets were used to make the proper names of gods and heroes fit into meter.
You're referring to divine epithets, and differences in cults are one part of it. There is a distinction between epithets used in literature, hymns, etc. and epithets 'guaranteed by their occurrence in liturgical formulae, dedications, and the official names of temples' (The Oxford Classical Dictionary). The OCD sums up this (most important, I'd say) class of epithets nicely: "(1) Purely local", which is what you pointed out: two cults may have two different epithets for the same god. "(2) Titles indicating association with another god" ... and "the largest and most important class of epithets", "(3) .... reference ... the functions of the god or goddess, either in general or with reference to some particular occasion on which his or her power was manifest." The best example of that is probably Zeus being referred to as the Thunderer.
HTH. And sorry I didn't use formatting.
In addition to what others have said, the gods filled various functions for the ancients. If you suffered some misfortune, you might know whom you've offended by their function. For example, if you're rude to a traveler and later misfortune befalls you, you might have offended Zeus because xenios is his domain. (In particular, xenios means "foreigner," so "Zeus Foreigner" as a name may refer to gods taking human form, a common motif in the ancient world e.g. Yahweh visits Abraham, or is just an abbreviation for "Zeus, Friend to Foreigners," or something like that.)