الرَّحْمَان Ar-Rahmaan (the Extremely Merciful), the Merciful One Who has as His attribute mercy, the One possessing tremendous and extensive mercy.
الرَّحِيم Ar-Raheem (the Bestower of mercy), the One Who has mercy upon the creation.
As the verifier has mentioned, the difference between Ar-Rahmaan and Ar-Raheem is Ar-Rahmaan is with regard to the attribute of Allaah’s self, the attribute He has Himself of extreme mercy and Ar-Raheem is His bestowing mercy upon the creation with regard to His action.
[Source: The Ninety-Nine Names of Allaah – Dawud Burbank [Audio|En] & Transcribed PDF]
ar-Rahmaan (The One Who is Most Merciful): This is one of Allaah’s particular names and which denotes the attribute of very great and extensive mercy that exists with Him.
[See Ibnul Qayyim’s Badaai’ Al-Fawaa’id (Beirut, Lebanon: Daarul Khayr, 1st. ed. 1414/1994)], vol. 1, pp. 23-24, and Ibn ‘Uthaymeen’s Sharhul ‘Aqeedah Al-Waasitiyyah, ibid.]
ar-Raheem (The One Who Bestows Mercy): This name denotes that the attribute is related to the marhoom (i.e. the one whom Allaah chooses to receive His Rahmah (Mercy).
So, the difference between ar-Rahmaan and ar-Raheem is that in the case of the former, ar-Rahmah is His attribute, and in the latter, ar-Rahmah is His act, meaning that He is the One Who bestows ar-Rahmah upon others. So, when both names come together [i.e. ar-Rahmaan, ar-Raheem], then the meaning in this context is that Allaah’s Mercy is very great and extensive, and that His Mercy reaches His creation. This is what is meant by those who say that ar-Rahmaan is the One Whose Mercy encompasses everything (general Mercy), and ar-Raheem is the One who has Mercy upon the believers (special Mercy). Since Allaah’s Mercy upon the kaafir is a special one in this life only, then it is as if they have no mercy upon them, because in the Hereafter when they will ask Allaah (سبحانه و تعالى) to save them from Hell and admit being wrong, Allaah will deal with them with His Justice, not His Mercy, Saying:
اخْسَئُوا فِيهَا وَلَا تُكَلِّمُونِ
“Remain you in it [Hell] with ignominy! And speak not to Me!”
[Qur’aan, Soorat Al-Mu’minoon (23:108)].
The name ar-Raheem is applied to Allaah (سبحانه و تعالى) and may be applied to others. Allaah (سبحانه و تعالى) has described His Messenger Muhammad (صَلَّى اللّٰهُ عَلَيْهِ وَسَلَّم) as Raheem (Soorat At-Tawbah, 9:128).
Ibn Katheer (رحمه االله) said:
“Some of the names of Allaah تعالى are restricted to Him only like the names: Allaah, ar-Rahmaan, al-Khaaliq, ar-Raaziq and so forth. Some other names, however, may be applied to other than Him.”
[See Badaai’ Al-Fawaa’id, p. 24, Sharhul ‘Aqeedah Al-Waasitiyyah, pp. 38-39, Lum’atul I’tiqaad, p. 29, Tafseer Ibn Katheer (1:21), Sharh Thalaathat Al-Usool within Shaykh Ibn ‘Uthaymeen’s Majmoo’ Fataawaa (Riyadh, KSA: Daar Ath-Thuraya, 1st. ed. 1414/1993)], vol. 6, p. 11.
Source : From the eBook: Tafseer Ayatul-Kursi – By Shaykh Muhammad bin Saalih Al-’Uthaymeen (rahimahullah). With Related Points of Benefit From the Shaykh’s Commentary on Imaam ibn Taymeeyah’s Sharhul ’Aqeedatil-Waasitiyyah, Prepared by Dr. Saleh As-Saleh (rahimahullah)
Related Link : https://abdurrahman.org/asma-wa-sifaat-com/