하고 같이 vs 와/과 같이
What is the difference between these forms? Do they both mean "together with", and if so, which is most frequently used? The sample sentence I have is:
아이들하고 같이 계세요?

Question

하고 같이 vs 와/과 같이
What is the difference between these forms? Do they both mean "together with", and if so, which is most frequently used? The sample sentence I have is: 
아이들하고 같이 계세요?

Answer

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

Answer

@JaeHyeok - I added the sentence to the OP

Answer

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.

Answer

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