One advantage of tools is that the cost of creating documentation about what you are doing is much easier, persistent, and accessible.
This can result in finer-grained tracking of the work that has been done, so that commits line up more or less exactly with items in the tracking tool. This can lead to a better accounting of what has been done and where the work came from (originally estimated and planned, discovered, added, etc.). With a physical board, the tendency is to tack on many commits onto one task, since there often just isn't enough room on the board.
If historical data is important to your organization (for estimation purposes, especially) the fact that the data may be accessible in a tool can be pretty helpful. If you notice that most of your work is stuff you hadn't even thought of then you know one area where you need to improve your estimation.
On the other hand, the usual things that people say about physical taskboards are true; they do promote discussions and visibility. But, I think that there are other ways to foster communication. I personally prefer tools because of the amount of historical data they provide.