We use “seem” when talking about a situation that we have experienced for a very short time and we want to give our impression. But, We use “sound” when we are listening to another person talking about a certain situation and we want to talk about our impression of it.

"Sounds like" and "seems like" are very similar. You could probably use them interchangeably. However, this is how I would describe their difference: "Seems like" means you SEE a situation and draw a conclusion. It seems like he is angry (from his behavior or the look or his face). It seems like it's going to rain. Look at those clouds. "Sounds like" means you have HEARD someone describe a situation and your are drawing a conclusion. So from what you just said, it sounds like your mom is not going to let you go. (I heard thunder in the distance.) Uh oh, it sounds like it's going to rain soon.