From experience I would say that there is more social interaction with board games. For instance, one video game that I have spent long hours playing with friends is Civilization 5. That game is a slow, turn-based game that takes hours to play (you build an empire from the ancient age to the modern age). Yet even when playing that slow-paced game, there are a lot of brain-bytes that are used trying to click on things and get all of your tasks done. I found that my friends and I would often just sit in silence while playing this game together online.
On the other hand, my friend and his wife often come over to our house, and my wife and I play Settlers of Catan with them. There is certainly a lot less 'things' to do while playing that game, and it is much less likely that we are going to sit at the table in silence while we play. A part of it very well might be that being with someone in person is much more likely to lead to conversation than 'talking' to someone over the internet through headsets. But that is just what I have experienced over the years.