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!"