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To some, these both words are same in meaning. Meaning merciful , Rehman and Raheem is one of the two attributes. Means mercy is pure caste. Some people have said that mercy means not punishing those who deserve it and do good to those who do not deserve good. In this case, the attribute of mercy will be functional ( If there is no creation, then Allah’s mercy cannot be manifested, but mercy will not exist at all ) Some people have differentiated between the two. Ar-Rahman is a form of exaggeration. The one whose mercy encompasses everything and the degree of Raheem is less than Rehman. ( The All-Merciful, the All-Compassionate. Merciful to him with a low degree of mercy) Some believe that Rehman is said to be in the sense that Allah is merciful to all His creatures. Believers or disbelievers, good or bad, Allah created them all and Allah provides sustenance. to all. Allah has said: My Mercy encompasses all things and the Most Merciful is said to be so, He is especially kind to the believers. Rehman is the special word but common in meaning and Raheem is the common word but special in meaning. 
The word Rehman does not apply to anyone other than Allah SWT that is why this word is special and since all creatures are created by Allah SWT and provides sustenance and Allah is the lord of every kind of gain and loss. In this sense, His mercy extends to all beings. It is because of this universality that he is called Rehman. The word Raheem is common, It also applies to others that is why this word is common but there is a feature in that which means special kindness and guidance. That is why this word is special because of its meaning. Ibn ‘Abbas said, “Both of these names are exact.”. Rehman is better than Rahim. Mujahid said: Allah is Rehman to the people of this world and Raheem to the people of the Hereafter, Ya Rehman Ad Dunya and Ya Raheem Ul Akhirah is said in Dua. Duhak said: Allah is Rehman to the people of heaven, He placed them in the sky and made obedience the shackles of their necks. Allah protected them from sins. And cut off from them all material needs and pleasures. And Allah is Raheem to the people of the earth. He sent messengers to them, and revealed books. Makkah Mukarramah said, “Allah is Rehman with one mercy.” Raheem with hundreds of blessings. 
Narrated by Abu Huraira The Messenger of Allah, may Allah bless him and grant him peace, said: Allah has a hundred blessings One of  his mercy, which Allah has sent down to the earth and distributed it among the creatures, People treat him with kindness and gentleness under the influence of mercy. Ninety-nine blessings he has reserved for himself. Who will descend on his servants on the day of resurrection. In other words, it has been narrated that Allah will complete it with them and will descend on His slave on the Day of Resurrection.
Rehman is the one who, when asked, will answer. And Raheem is the one who is angry if his servant is not asked. This is an accident narrated by Abu Hurayrah. The Messenger of Allah said: He who does not ask Allah SWT. Allah SWT is angry with him. There is a saying of a poet that if you do not ask to Allah, then Allah SWY will be angry and if you ask man, then also Allah SWT will be angry. Allah is Rehman, He has bestowed blessings and bounties. Allah is Raheem he stopped and removed the pain. Allah is Rehman he save us from hell and said: You were on the edge of the pit of fire. Allah saved you from it. Allah is Raheem and will enter Paradise. He said, “Enter Paradise in peace without fear.” Allah is Rehman, His mercy is on the souls. Allah is Raheem, His mercy is on the hearts.
Allah is Rehman he removes difficulties, He forgive all sins with Raheem. Allah is Rehman, He explains the right path. He is Raheem, He save from wrong path, And gives way to the path of righteousness. Allah is Rehman, He forgive all sins even it is equal to the sea. Allah is Raheem, He accepts goodness Although not pure. Allah is Rehman, He gives the means to correct the livelihood. He is Raheem, Rehman is the one who shows mercy. He take away pain, He has the power to repel evil. Allah is Raheem, Raheem is the one who provides Sustenance and food. It is a fact that Allah is the Sustainer and the Strong. Allah is Rehman to His disbelievers. He is the Raheem to the people of Tawheed.   He is Rehman for the ungrateful people. He is the Raheem for those who give thanks to Allah. He is Rehman is for those who associate it with Him,  And Raheem to those who call Him One.