In negative sentences, they are exactly the same.
"I don't like anyone other than John"=="I don't like anyone besides John"=="In the entire world, I only like one person, and that person is John."
In positive sentences, "other than" has a narrower and more precise definition than "besides".
"Other than John, I like everyone here"="I like everyone here except John. I hate John."
"Besides John, I like everyone here" is ambiguous. It might mean the same thing as the "other than" sentence, or it might mean "I like John, and in addition I like everyone else here, too." In positive sentences, you need to use context to figure out which of these meanings is intended.