There is only a minor difference which has to do with the respective object.
These two words are often mistranslated and mixed up.
The two words al-Rahmān (الرَّحْمٰن) and al-Rahīm (الرَّحِيم) look almost the same.
The Arabic root r-h-m (ر-ح-م) means: to have mercy (upon s.o.); to have compassion. The dictionary by Hans Wehr translates both words with merciful.
However, there is a difference in meaning.
- Al-Rahmān is more intensive, including in its objects the believer and the unbeliever, and may be rendered as The Compassionate. Or: The Most Merciful.
- Al-Rahīm has for its object the believer onlyand may be rendered as The Merciful. Or: the Most Beneficent [to the believers].