We also use waiter/waitress, maître d', and sometimes steward/stewardess (esp. on airplanes). There's technically a difference:
- waiter/waitress - brings food
- server - person the customer interacts with (i.e. takes orders)
- maître d' (hotel) - head of wait staff
- steward/stewardess - serving customers is usually a secondary duty
I think "server" has become more popular because it's gender neutral, but "waiter/waitress" is still quite common and most don't make the distinction between the two.
I personally like the overlap between computer server and restaurant server because both exist to provide things upon request. The term "wait" that "waiter" comes from is pretty archaic.