"처럼" and "같이"
When they are used in the sense of "like (something)" after a (pro)noun, they are completely interchangeable.
The difference is that 같이 has the following additional meanings and usages.
1. As a particle, 같이 sometimes conveys a special emphasis instead of "like". 처럼 may or may not work.
- 나는 새벽같이 떠났다. (I left brightening early) - 새벽처럼 is unnatural.
- 쟤는 매일같이 저런다. (He does that day in and day out) - 매일처럼 can work as a poor second choice.
2. For "like" or "accordingly", 같이 can be used as an adverb while 처럼 is always a particle coming after a noun.
In such usages, 같이 comes from the adjective 같다 (is the same). 처럼 has no other related word.
- 선생님이 하는 것처럼 하세요 / 선생님이 하는 것같이 하세요. (... like I do)
- 선생님이 하는 것과 같이 하세요. (This is more clear as it emphasizes "THE WAY AS I do")
(For emphasis, you can also use 똑같이, which is THE EXACT SAME WAY AS ...)
3. 같이 also means "together" or "with someone".
- 나하고 같이 여행 갈래? (Would you like to travel someplace with me?)
- 우리 점심 같이 먹자. (Let's have lunch together)