Kids' b-days aren't that formal, but I do paper invites for DD#1's parties with a regular RSVP (phone or email) included so I can get a headcount, since we make the food and I need to know how much to buy/cook. I don't think it would make a difference, but I think the paper ones are cuter :)
For "events" (get togethers) for an adult birthday, a facebook "event" or an evite are both fine with me. But a "Hey everyone, come to this," wouldn't work. It would have to be people specifically selected for the event (not a whole friends list, for example), mostly because (a) I don't want all those people coming, and (b) then I have only a few to follow up with--and in a worst case scenario, I have a top number of potential guests (that is, 16 invited, 12 yes, 2 no, 2 no answer--I can tell a restaurant, say, 14--since those 2 may just show up anyway.)