They are similar but not exactly the same. The sentence I am afraid to say indicates that you do not want to say something because of fear of what will happen if you do so. While the sentence I regret to say indicates that you are goint to tell the other person something that you are not happy to inform but you will say it anyway. I hope this helps.

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 ...'

They are both polite expressions. I regret to say sounds more serious than to say I'm afraid to say