Very similar phrases, both are comparisons. I strikes me that the first sentence is based on quantity and the second based on quality. I like you the most - more than any other i.e. more than/less than I like you best - better than the rest i.e. good/better/best

I think they are interchangeable.

"I like you the most" can be used by itself, or can be put into a 'selection' context like: "out of all the teachers, I like you the most". "I like you best" does not make sense by itself. Rather, it needs to be qualified further like: "I like you best when you're not angry."

i like you the most = upon everything or everybody else i like you best = upon the ugly neighbor ( just kidding, upon a specific person)