Ar-Raheem

Ar-Raheem and Ar-Rahmaan comes from the same root. Thus, in principle, there is no difference in meaning except that Ar-Rahmaan is a bit more vast and more general than Ar-Raheem in its sense. That is, Ar-Raheem is a little more restricted compared to Ar-Rahmaan which means a being who has a very particular and special grace for the believers in the Aakhirah only. As for Ar-Rahmaan, it means a being who has pity for all His servants, believers and non-believers. Thus, when we say: بسم الله الرحمن الرحيم, it means "In the name of Allah who has mercy on all His creatures without distinction and who will have a special pity for the believers in the Aakhirah."

In the beginning of the Qur'aan-e-Kareem, Allah Ta’ala introduced Himself to His servants as الرحيم الرحمن to show that He is filled with grace and immense pity although He is the owner of the universe.

Therefore, there is no failure in obeying His commandments. What a person may expect from a boss who is very kind and generous, even  more should be expected from Allah Ta’ala. 

Once Hazrat Aa'isha (Radiyallahu Anha) read:

"...اللهم انى ادعوك الله وادعوك الرحمن وادعوك الرحيم"

Rasulullah (Sallallaahu Alaihi Wasallam) had heard that and informed Hazrat Aisha (Radiyallahu Anhu) that Al-Ism-ul-A'zam is found in those names that she has quoted / cited. (Ibn Majah)

[English version of "Ar-Raheem" – Al Jamiat – Vol.3 No.13]