Al-Rahmaan and al-Raheem are two of the names of Allaah which refer to Allaah’s attribute of Mercy.
Al-Rahmaan refers to the vastness of Allaah’s mercy, and al-Raheem refers to its effect on His creation. So al-Rahmaan is the Owner of vast Mercy, and al-Raheem is the Owner of Mercy that encompasses His creation.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: “Al-Rahmaan is the Owner of vast mercy, because the fa’laan form in Arabic indicates vastness and abundance, as it is said rajal ghadbaan (a very angry man) when he is filled with anger.
Al-Raheem is a name which refers to the action, as the fa’eel form refers to the doer of an action. So the phrase al-Rahmaan al-Raheem indicates that the mercy of Allaah is vast, as is understood from the name al-Rahmaan, and that it encompasses His creation, as is understood from the name al-Raheem. This is what some of them meant when they said that al-Rahmaan refers to mercy in a general sense and al-Raheem refers to mercy that is specifically for the believers. But what we have mentioned is more accurate.”
From Sharh al-‘Aqeedah al-Waasitiyyah, 1/22
And Allaah knows best.