Natalie, both have similar meanings. In the US, we say "get along with." "Get on well with" is a British phrase, and isn't used here often.

'get on well with' is more positive, and would refer to a slightly better relationship than 'get along with', particularly if you are using both phrases. It can be used to describe any social relationship, and even extends to groups. "I get on well with my mothers new husband". "My cat and dog get along with each other pretty well". "the Maori wardens get on well with the local police".