같이 and 함께 both mean together but 함께 sounds more formal.
EXAMPLE:
같이 가요. - Let's go together.함께 가요. - Let's go together. (more formal)
So you won't hear people saying 함께 that much in everyday conversations with friends.
MORE EXAMPLES:같이 영화 볼래요? - Do you want to watch a movie together?
함께 영화 볼래요? - Do you want to watch a movie together? (more formal)
Because of this difference in formality, when you want to say together with someone, the word with is more commonly expressed through -(이)랑 when you say 같이 and through -와/과 when you say 함께.
EXAMPLE:
우리랑 같이 해요. - Do it together with us. (more casual)
저희와 함께 하시죠. - Do it together with us. (more formal)

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KOREAN VOCABULARY LEARNING
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