Generically speaking, "sorry" carries certain degrees of regret or apology. If I step on your shoe, I will say " I am sorry ....", I will not say however, " I am sad". If Mary found out her boyfriend has been two-timing, you may say two her: " I am sorry to hear that. Don't be sad. Be happy that you find out now than later. It is not natural to say " I am / feel sad for you..... ".