They are suggestions: Would you like to have a bite to eat? Where would you like to eat? Answer with your druthers, real or otherwise: Yes, I would. How about McDonald's?
My friend said that "Would you like to?" sounds a little too formal on certain occasions. For something that a decision should be made immediately on "Do you want..." sounds better.
I think that "Do you want..." does not give a suggestion, instead, the speaker tries to find out what the person is going to do, i.e., the person's decision. I'd like to hear more from
English seldom sounds 'too' polite. I think 'Would you like to?' works fine in almost any situation-- I certainly use it to my wife and son when I am being nice.
'Do you want to?' is the familiar equivalent, used only with wives and sons and other relatives, close peers, and close friends and acquaintances.
They are both suggestion forms, though of course '