for example, Let's go together : 같이 가자. Let's go separately : 따로 가자.

같이 is more frequent used expressin to speak and 함께 is usually used in writing, in other words, a literary style. The meaning is Same.

with= 와/과 같이/함께 without=이/가 없이 for example: 우리는 선생님과 같이 갔어요.(we went with our teacher) 우리는 선생님이 없이 갔어요.(we went without our teacher)