1. same meaning I think perfectly same. 선생님(과 같이) 가다 = 선생님(하고 같이) 가다. 2. "와/과 같이" could be used as "~처럼". "~처럼" means "like" For example, "그는 아이(와 같이) 순수하다" is same meaning "그는 아이(처럼) 순수하다" He is pure "like" child

@JaeHyeok - I added the sentence to the OP

Ah... both of them mean "together with" but it is a part of meaning. '같이'(as a adverb) means 1. together with 나 하고 같이 해요. 친구와 같이 해요, 2. likewise, same, similary 내가 하는 것과 같이 하세요= 내가 하는 것처럼 하세요. Could you give me the sentence that made you get confused? It is hard to explain because I am a not a Korean teacher. The sentence's context can help me.

하고 같이 is said rather than written. 와/과 같이 is written rather than said. and ~랑 같이 is said rather than written, and more used than 하고 같이