بسم الله الرحمن الرحيم
Sheikh Muhammad ibn Salih Al Uthaymin said,
Ar-Rahman: The possessor of vast mercy.
Ar-Raheem: The granter of mercy to whomever He wills.
The difference between the two is that the first relates to the existence of mercy as a description for Him, whereas the second is related to it being an action of His, as He grants (or bestows) mercy to whomever He wills from His creation. [شرح لمعة الاعتقاد]
Sheikh Abdur Razzaq ibn Abdil Muhsin Al Abbad said,
Ar-Rahman: He to whom which mercy is a description.
Ar-Raheem: The doer of the action of being merciful to His servants, for this reason the Exalted says:
“And with the believers He is most merciful (رحيمًا) ” [Al-Ahzab: 43]
“..certainly He is to them full of Kindness, Most Merciful (رحيمٌ)” [At-Taubah: 117]
So Ar-Rahman is on a (certain) scale (in the Arabic language related to morphology) that indicates a permanent, inseparable, and perfect attribute. In other words; from His attributes is mercy. Whereas Ar-Raheem indicates the spreading of mercy to those whom it is bestowed upon. Meaning those who received mercy via (His) action. [فقه الأسماء الحسنى]
Compiled by Najeeb ibn Yusuf Al Anjelesi.