I posed this question to a friend of mine in Italy who is a native speaker. I was curious myself about the difference. Her answer was, 'We used to say "tavolo" when we were talking about a table or even a desk, any table no matter what, but now when we talk specifically about the table where we eat, whether it be in the kitchen or the living room, then that same table for us becomes "tavola.'
I received my answer from my friend in Italy. Tavolo/tavoli is always table/tables. The plural of tavola IS tavole, but depending on the context of the sentence, these words can refer to either a table or a plank of wood as those used in construction.