Bismillah.
Ar-Rahman: The Entirely Merciful, Ar-Raheem: The Especially Merciful
Both of the names Rahman and Raheem come from the root Ra-ha-meem (ر ح م), these names are similar so they require a parallel explanation. What is Rahmah? Rahmah is mercy. In the dictionary it means: compassion, kindness and gentleness. Also when a person is kind, compassionate and tenderhearted towards the other. So Ar-Rahman and Ar-Raheem: The One who has a lot of Rahmah, The One who shows a lot of rahmah, He expresses a lot of Rahmah. There are two Names of Allah with the same root, to emphasize that Allah is really the MOST Merciful, there is no one else who is more merciful than Allah subhanahu wa ta’ala.
Ar-Rahman is structure of fa’laan, Ar-Raheem is structure of fa’eel. They both have mubaalaghah (intensification). What is the difference? Rahman has more mubaalagah (intensification) than Raheem. So ar Rahmaan indicates the One who has extensive and great mercy. Who is Ar Raheem then? He is the One who has extensive and great mercy but He expresses it on the ones whom He chooses. So Ar-Rahman extends to all of His creation including disbelievers, and Ar-Raheem extends to Allah’s special mercy only for the believers.
Examples from the Qur’an
20: 5->Ar Rahman ‘alal ‘arsh istawaa’. The Most Merciful, upon the throne He rose. He established His throne and showed Mercy towards all His creation.
55:1-5-> Ar Rahman, Who is He? ‘alamal Qur’an, He taught the Qur’an, khalaqal insaan and He created mankind. This mercy is not just for the believers, but for everyone (insaan).
Raheem only to the believers: 33:43-> wa kaana bil mu’mineena raheema. One who is especially merciful to the believers.
2:37-> Innahu huwa at-Tawwaabur Raheem. Indeed He is the Greatest Acceptor of Repentance, the especially Merciful. Tawwaab is for those who turn to Allah, ie the believers.
Combinations of these Names:
These Names appear many times together. First example, the basmalah, and second verse of Fatihah. They appear together to express a lot of Mercy, Allah is the One who is THE MOST Merciful.
When we look at Ar-Raheem, it appears with other Names:
At-Tawwaabur Raheem, Ar-Raoufur Raheem, Ghafoorur Raheem, Al-Azeezur Raheem (surah Shu’ara). Remember His mercy, but also remember His might.
Ar-Rahman
Appeared many times in surah Maryam and Furqan. Why so many times? These surahs were makkee surahs and they were introductions to the deen, so one of the first ways of how Allah Introduces Himself is by which attribute? The attribute of Rahmah, which is why the Qur’an opens up with these Attributes as well (Basmalah and Faatihah). Why, what’s so important about them? First of all, when people learn about Allah, they shouldn’t learn He is the One who puts them into hellfire but that Allah is Ready to accept people and forgive people. This is why messengers are basheerun (giver of glad tidings) and natheer (warner). Of the first ways to introduce Allah ta’ala in da’wah is by His Mercy. Specifically for the believers, they should remember His Mercy then they should abstain from wrong actions. Think: ‘If Allah is SO Merciful to me, why should I disobey Him and do wrong? Allah is so Rahman and I am so disobedient.’ The siffah (characteristic) of Rahman and Raheem remind the believers that they should remain firm on Allah’s Obedience. Allah is So Merciful, So kind towards you, so why won’t you obey Him?
A Hadeeth in Bukhari states: “Verily Allah created Mercy. The day He created it, He made it into one hundred parts. He withheld with Him ninety-nine parts, and sent its one part to all His creatures.” Allah has saved the 99 parts for His true believers on the day of judgment, that is when Allah will be ar Raheem. Allah will say to the disbelievers: do not even speak to Me, remain in hell-fire (23:108). Why? Because they do not deserve the rahmah of Allah. So if a person does not not benefit from His Mercy in this world, they do not receive in the aakhirah. To benefit is not to just receive it in this world, but to accept the guidance and stay away from disobedience.