This article has some nice info: http://blogs.transparent.com/italian/tavolo-or-tavola/

For completeness, though, the difference is a little harder to pin down: for instance "mettere le carte in tavola" (to reveal all your plans) has nothing to do with the dining room but rather with playing cards. The dictionary has a lot of entries for both words.

Abbiano is subjunctive, which is an unreal (irrealis) mood; in that case you wouldn't be sure that they have a table. It's mostly used in subordinate sentences, e.g. "I believe that they have a table" -> "(io) credo che (loro) abbiano un tavolo".