"Rahman" means meaciful for all (believer or non-believer). like sending rain, food,...

"Rahim" means merciful for believers only. 

Allah has two kind of mercy. one is for all. and one type is extra only for believers (Mumenin).

[Source][1]


  [1]: http://www.shiasource.com/al-mizan/tafsir/1-1-5/