For your information, a ‘get-together’ is an informal social gathering. At Christmas time many families in our neighborhood host a get-together. Coming together often has a purpose. We came together to do something.

In certain situations they are almost the same, people joining/meeting each other 'My whole family gets together for dinner every Christmas.' 'My whole family comes together for dinner every Christmas.' However, come together can be used more generally, and to focus on togetherness/unity. 'The neighbourhood came together to clean up the street after the festival.' Here the emphasis is that the people in the neighbourhood are working together, not just meeting.