Ar-Rahman signifies Allah, the Glorified's, attribute, while ‘Ar-Raheem’ signifies its relation with the object of mercy.
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So the prior is for a description and the latter is for an action.
The prior signifies that Mercy is His attribute, while the latter signifies that He is merciful towards His creation with His Mercy.

To understand this, contemplate His saying
'Ar-Rahman' followed by 'with them/to them' never comes in the Quran[1]. So it becomes known that ar-Rahman is: the One who is attributed with mercy, and Raheem is: the One who applies mercy upon His creation.
'Ar-Rahman' - He is attributed with vast mercy because this word form in the Arabic language signifies vastness and fullness.

'Ar-Raheem' – a noun that signifies the action[2].

When combined, 'Ar-Rahman' and 'Ar-Raheem' mean that Allah’s mercy is vast, and reaches the creation. This is what some scholars alluded to by saying:

'Ar-Rahman' is a universal mercy [to all the creation] and 'Ar-Raheem' is a specific mercy for the believers.

Since Allah’s mercy upon disbelievers is specific to the life of this world only, it is as if there is no mercy upon them, because in the afterlife, when they ask Allah to remove them from the Hellfire, and seek intercession with Allah through His Lordship, and confess against themselves saying: The name Ar-Raheem (in Arabic: ٱلْرَّحِيْمُ) comes from the same root as Ar-Rahman which refers to Allah’s attribute of being Merciful. Although similarly related the meanings are different. Ar-Rahman can be understood to refer to Allah as the origination/source of all mercy, whereas Ar-Raheem refers to its sustaining/infiniteness. Some view the name of Ar-Raheem as being merciful to His creation and deserving of mercy (primarily the mercy reserved for the believers)