I'm with you on words just being words. But there is one factor, especially in ongoing relationships (romantic, platonic, familial, whatever), that makes apologies important.
I don't care about the feeling bad or the words or whatever, but I do care about acknowledging wrong-doing. Because if they haven't apologized, they might not think what they did was wrong, and there is absolutely no reason to believe if the situation came up that they wouldn't do it again.
Now, don't get me wrong, plenty of people apologize and then do the thing again. But if they don't even apologize, it's practically guaranteed. The apology just sends the message that you and they are on the same page about whatever it is.