Both "tidy" and "neat" mean arranging things in order. "Tidy" has more word forms comparing to "neat". "Tidy" can be used as an adjective and a verb; while "neat" is usually an adjective. Here are some examples: 

Tidy (adj) - Your table is very clean and tidy. 
Tidy (noun) - I finally have time to tidy my room. 

Neat (adj) - The latest magazines are in the neat pile. 

A fun fact for you: Neat can also means good in informal conversation. Here are some examples: 

1. I took an English on italki and the teacher was neat! 

2. "How's your day?"
"I went to a concert with my friends and I took a selfie with some celebrities!"
"Neat!"