'Befriend' sounds quite formal and old-fashioned, and is not very often used. It's also worth remembering that 'befriend' is a transitive verb, so it sounds rather one-sided. If you say 'Ann made friends with Maria' or 'Ann and Maria became friends', this sounds like a mutual arrangement. However, if you say 'Ann befriended Maria', the action is only Ann's. It suggests that Maria was a lonely person whom Ann befriended out out kindness or charity.

There is never a time where one MUST be used instead of the other that I can think of, but there are a few connotative differences between them. "Befriend", to me, is more formal, and I would also be more likely to use it when talking about animals. (E.g. "I befriended a dog yesterday.") "Make friends with" is the opposite: less formal, with me being more likely to use it when talking about a person. Overall, however, the difference is slight - saying "I made friends with a dog yesterday." sounds perfectly natural as well. If there is a difference you should heed, it is the formality.

To me they are synonyms. To "befriend" sounds more formal, so I would normally say "make friends with" in everyday speech.