How vital is alignment? And more importantly, is there any benefit/reason for being neutral? The in-game help said that some items will only be equippable for certain alignments. So, if I run neutral, am I missing out? The reason I ask is that some of the gray Jedi have been my favorites, but I'd hate to see them at a disadvantage here.

And then back to the first question. Does alignment factor into the story or quests/decisions any later on? I don't need spoilers, just wondering if it's something I should think of beyond just wanting to wear some cool Sith robe or something. For instance, if I pick Republic early on, but go way dark side, am I still with the Republic? So, I basically want to know if there are any consequences of my alignment later on, other than what I can equip.

At the moment, yes, you are missing out if you stay neutral/gray. Bioware hasn't put in the "gray" armor/items yet, so you'll likely be missing out on some of the better armor you could get if you chose one of the other alignments.

That said, there is supposed to eventually be good armor for folks that go gray. Whether you want to wait for it or not, it's hard to say.

You are still aligned with your faction, no matter what morality you follow. That much is consistent. I'm not sure if actual story is affected, but your companions can and will be.

The only current 'benefit' to choosing one over the other is appearances; there's no guarantee it'll stay purely cosmetic or if they've more long-term plans with it.