Sometime they could be synonyms, e.g. "Puso dinero en su bolsillo" o "Metió dinero en su bolsillo". Both are ok. However, there are other expresions that would make no sense: "Se puso la ropa" --> Ok... "Se metió la ropa"--> it does not exist.

Meter-> to place something INSIDE of something else. Poner-> the closest trasnlation is "to put".