Shaykh Hasan Ali gives a comprehensive lecture on the Names of Allah - Ar Rahman and Ar Raheem. They are from the same root or etymology - Rahma. The Mercy of Allah covers everyone and everything whether its a Mu'min or a Kafir and every creation. He has a Greatness to His Mercy which is awe inspiring and there is no end to His Mercy
Ar Rahman is that Name of Allah which is directed for His entire creation - believers and non believers whereas Ar Raheem is that Name of Allah that is specifically meant for those Mu'mins or believers on which Allah has His Rahma. For example, in Surah Al Ahzab ,verse 43, Allah says,
It is He who confers blessing upon you, and His angels [ask Him to do so] that He may bring you out from darknesses into the light. And ever is He, to the believers, Merciful.
Imam Al Qurtubi gives numerous examples from the Qur'an where these names have been used:
Surah Al Fatihah - [All] praise is [due] to Allah, Lord of the worlds. The Entirely Merciful, the Especially Merciful.
Surah An Naml, ayat 30 - Indeed, it is from Solomon, and indeed, it reads: 'In the name of Allah, the Entirely Merciful, the Especially Merciful,
Surah Al Baqarah, ayat 163 - And your god is one God. There is no deity [worthy of worship] except Him, the Entirely Merciful, the Especially Merciful.
Surah Ar Rahman, ayat 1 - The Most Merciful
Surah At Taubah, ayat 117 - Allah has already forgiven the Prophet and the Muhajireen and the Ansar who followed him in the hour of difficulty after the hearts of a party of them had almost inclined [to doubt], and then He forgave them. Indeed, He was to them Kind and Merciful.
The name Ar Rahman can only be used for Allah and it cannot be used for any other human beings. Ar Rahman denotes the Mercy of Allah for everyone like rain, our existence, food, water etc . Ar Raheem denotes the Mercy of Allah specifically for Muslims such as His Hidayah for us to be Muslims with Iman. Imam Al Qurtubi says that Allah has given us the possibility to call ourselves Ar Rahim if such a person possesses merciful traits.
Effects of the Mercy of Allah as illustrated in the Qur'an :
Surah Al Qasas, ayat 73 - And out of His mercy He made for you the night and the day that you may rest therein and [by day] seek from His bounty and [that] perhaps you will be grateful.
Surah Al Ankabut, ayat 21 - He punishes whom He wills and has mercy upon whom He wills, and to Him you will be returned.
Surah Al Anbiya, ayat 107- And We have not sent you, [O Muhammad], except as a mercy to the worlds.
Imam Al Qurtubi also mentions that Allah has told us to never lose hope of the Mercy of Allah and also to not lose fear of Allah in Surah Az Zumar, ayat 53:
"Do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.”
Also, in Surah Al Hijr, ayah 56 - He said, "And who despairs of the mercy of his Lord except for those astray?"
Also, in Surah Al Ankabut, ayat 23 - And the ones who disbelieve in the signs of Allah and the meeting with Him - "Those have despaired of My mercy, and they will have a painful punishment".