In sentences like "I like this one the best" or "I think that one fits me the most" you can use "most" and "best" interchangeably, but in most cases you can't.
In the sentence "He is the most handsome", "most" intensifies the adjective "handsome." No one is more handsome than him. Most doesn't necessarily have to be positive. You could also say "he is the most hideous." The word "best" cannot be used in either sentence.
"Best" is an adjective, and it essentially means "the most good". "Best" modifies nouns. "He is the best athlete". "She is my best friend." You can't use the word" most" in either sentences.
Both "the most" and "the best" can be used as adverbs, too, as in the sentences you provided. In most cases they mean different things though;
"She runs the most" means she runs more often than the others.
"She runs the best" means she is better at running than te others.