It's actually more complicated than it seems. Well is an adverb, but can also be used as an adjective (as when it is a predicate adjective).
We say, "The band sounds good."
But we also say, "The band plays well."
The difference has to do with subtle differences between sounds and plays. Sounds is a linking verb and needs something to follow it. Plays is not a linking verb. You could say "The band plays" and that would be a complete sentence. You can't say, "The band sounds."
Well also has special meaning when referring to the four senses.
Spot smells good. (Tells us that Spot has had a bath. Smells is a linking verb here. Focus is on Spot, not his ability to smell, and doesn't refer to his health.)
Spot smells well. (Focus is on Spot and tells us that Spot doesn't smell like he is sick.)
Spot smells well. (Focus is on "smells" and tells us that Spot is a bird dog.)
Me and camden love this stuff.