‘Ar-Rahman signifies Allah, the Glorified's, attribute, while ‘Ar-Raheem’ signifies its relation with the object of mercy.
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:
And He is ever, to the believers, Merciful. Al-Ahzab: 43
Indeed, He was Kind and Merciful to them. At-Tawba: 117
'Ar-Rahman' followed by 'with them/to them' never comes in the Quran. 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.
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 really 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:
Our Lord! Take us out of it, if ever we return (to evil), then indeed we are oppressors Al-Muminoon: 107
They will not find mercy, rather they will find justice, Allah – the Mighty and Majestic - will say to them:
Remain in it in humiliation! And speak not to Me! Al-Muminoon: 108
Sharh al-'Aqeedah al-Wasitiyyah: p.28
 This demonstrates a highly in-depth analysis/contemplation of the Quran where the Imam went through every Ayah where Ar-Rahman and Ar-Raheem were mentioned, and compared the two to reach this conclusion.
 Additional linguistic benefits were removed but can be found in the arabic version