There is very little difference between these two expressions. They are both used as a way of introducing a fact that the listener may find distressing or unhelpful. Contrary to what Sergio suggests, 'I'm afraid' has nothing to do with fear. The meaning of 'I'm afraid' here is the same as 'I regret'. Regret is more formal, but they are used in much the same way.
In essence, they both mean 'I'm sorry, but ...'