We use both words at same meaning. You don't need to find a difference between "함께", "같이". They are same, means "together(with)"
But They means a little bit different.
같이: use for two more people or things
ex) 1.친구와 같이 사업을 하다. Make a business with my friend together.
2.모두 같이 갑시다. Let's go together.
함께: together at once (※ 함께 usally use when you want to say "together with somebody" )
ex)1. 온가족이 함께 여행을 갑니다. All my family go travel together.
2. 나는 동생과 함께 공부합니다. I study with my brother together.
I wish my answer would help for your question.