Typically "I'm sorry you feel that way" is used when you don't feel that you are in the wrong. It is more of an olive branch, or an agree to disagree but have regretful feelings about it.
The "agree to disagree" thing is at best condescending, at worst a desperate way to escape a thorough discussion where you're pretty sure you're going to "lose" i.e. have to face an uncomfortable truth to begin with.
"I'm sorry you feel that way" is a cop out and a non-excuse. It makes the sender feel better and the receiver feel worse.
Well, I'm sorry that you feel that way.
It seems that we have a differing opinion on the subject but I still value your opinion and we are still friends.
See how that works? I still think I am right, you still think you are right. I wish I could bring you over to my side but it seems that I can not so I regret your feelings. Then I told you so, (olive branch).
Good talk bro. :)
:) let's see.