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.

They are the same in meaning. In daily conversation, 같이 is much more common. 함께 is used mostly in song and poetry lyrics and formal writing.