My understanding of the difference in the words is this: unreal is the easiest to comprehend. It’s just the opposite of real. So, it’s like not real, or not comporting with reality. „He has unreal expectations of making that amount of money at his first job.“ (the expectations have no basis in reality) 

surreal is an odd word that is used far less commonly than unreal. It refers mostly to a person‘s feelings, experiences or perceptions. Or, I’ve heard it used to describe artwork. Examples: „That experience I had was surreal.“ This would mean something like a very strange experience, one that was so out of the ordinary,  it seemed out of this world, or too odd to be real. But, it really happened. With artwork you might say the painting was surreal. To me that means the painting is abstract,  not like a painting of a horse standing next to a barn.  

"She had unreal expectations of her new job."

Surreal means something that is bizarre.

"It was a surreal experience walking in Tokyo with not a soul in sight."