How the attributes of god 'Rehman' and 'Raheem' different

Question

Except surah 'tauba' all surah of Qur'an starts with "bismillah ar rehman ir Raheem".
How being rehmaan and being Raheem different?

An aalim explained that Allah is rehmaan for all the creatures. It does not matter if they believe in god or not, it does not matter they are good or bad.

And Allah is Raheem for some selected people.

Answer 1

"Rahman" means merciful for all (believer or non-believer). Like sending rain, food,...

"Rahim" means merciful for believers only.

Allah has two kind of mercy. One is for all. And one type is extra only for believers (Mumenin).

Source

Answer 2

Some aalims (scholars) say: "Rahman" is a name ("Esm") not attribute, just for god. But "Rahim" is an attribute which used for passenger "Mohammad (ص)" in surah "tauba".
Two reasons are explained:

1. We can replace "Allah" instead of "Rahman" everywhere in the Quran.
2. "Rahim" used just in kindness of him, but "Rahman" used everywhere.

From here.

Answer 3

Both words are primarily adjectives. Rahman is used both as a proper noun for God (e.g. 17:110) as well as an adjective (e.g. 59:22):

Since you asked this question in reference to Bismillah, it should be noted that both are used as attributes here. Both words allude to different shades of meaning of رحمة (kindness):

1. Rahman indicates the high intensity of رحمة and
2. Rahim refers to its perpetuity.

Let me quote Islahi. He writes:

The noun Rahman is the intensive form of Fa‘lan (eg: Sakran and Gadban), while the noun Rahim is an adjective of the form Fa‘il (eg: ‘Alim and Karim). A look at the usage of the Arabic language shows that the form Fa‘lan expresses great fervency and enthusiasm, while the form Fa‘il expresses steadiness and perpetuity. In other words, the first depicts vigour and the second constancy in God’s mercy. A little deliberation shows that the Almighty’s mercy on His creation possesses both these characteristics. The enthusiasm and warmth is complemented by permanence. It is not that His attribute of Rahman induced Him to create, and He later forgot to foster and sustain His creation. Indeed, He is nourishing and taking proper care of them because He is Rahim as well. Whenever a person invokes His help, He hears his calls and accepts his prayers. Also, His blessings are not confined to this world only. Those who lead their lives according to the path prescribed by Him, shall be blessed with eternal life and joy. It must be conceded that all these aspects cannot be comprehended without an integrated understanding of these attributes.