I think " you sound ... " mean as far as I see you are like me ( not sure just seem ) ....but in "you are like me " there is no doubt

So if someone says something you usually say its you sound like me whereas if someone does things similar to you they would be like you

You are like me means we think the same way about things or have the same attitudes. You sound like me means either we sound the same or we speak the same way