al-Salāmu ‘alaykum wa rahmatullāh,
Undoubtedly the most oft-repeated names of Allāh are al-Rahmān الرحمن and al-Rahīm الرحيم, due to them being present in the basmala  which is mentioned before every chapter in the Qur'ān and which muslims are instructed to repeat before beginning any task.
For two names to so constantly be mentioned alongside the name of Allāh alludes to their status and importance within the Islamic creed, and thus it is important to gain a thorough understanding of their meaning and significance.
Both al-Rahmān and al-Rahīm are derived from the root rā' - hā' - mīm (ر-ح-م) and mean to treat or regard someone with mercy, compassion or tenderness. From the same root stems the word al-rahim الرَّحِم (the womb) for the womb itself can be seen to behave in a tender and compassionate manner towards the fetus which it carries.
While both words are derived from the same root, the difference in their meaning lies in their form. Rahmān is a hyperbolic form (sīghah mubālaghah) and conveys the meaning of extensiveness and endlessness. It is a quality that is inherent in and inseparable from Allāh. Rahīm is a permanent adjective (sifah mushabbahah) and conveys the meaning of constant renewal, and of a quality issued forth as necessitated by the actions and behaviour of the recipient.
In addition, the term al-Rahmān is applicable only to God - even in Pre-Islamic times the Arabs did not refer to a man as being al-Rahmān, but rahīm. The only recorded instance was that of Musaylimah the Liar who was dubbed Rahmān al-Yamamah, but even then not 'Rahmān' alone. On the other hand, rahīm can and always has been used to describe humans - the Prophet himself salla Allahu 'alayhi wa sallam - was described in the Qur'ān as being:
بالْمُؤْمِنِينَ رَءُوفٌ رَّحِيمٌ
to the believers is kind and merciful.
But why are two words that denote similar meanings used together in the same clause? The answer may lie in how the sequence of Allāh's Names in the basmala graduate in two opposing but harmonious ways;
- general to specific: the absolute pinnacle of mercy in every form - conceivable or not - that is encompassed in the Name Allāh, followed by mercy being extended to everyone unconditionally (as denoted by al-Rahmān), followed by mercy being extended as a response and reward for specific actions (as denoted by al-Rahīm).
- specific to general: the application of the names themselves: Allāh is the Creator and can refer to none but Him - it cannot be used as a description but can only be described (one can not say, for example al-Rahmān, Allāh where al-Rahmān is the noun and is the adjective. This is followed by al-Rahmān which is an adjective that can only be used to describe Allāh, followed by Rahīm which is an adjective that can also be used to describe people.
Perhaps if this angle is recalled when repeating the basmalah we would be able to more completely grasp the totality of mercy indicated in His saying,
وَرَحْمَتِي وَسِعَتْ كُلَّ شَيْءٍ
My mercy encompasses all things
 The basmala is an example of naht used to refer to the saying of bism Allāh al-Rahmān al-Rahīm (In the Name of Allāh, al-Rahmān, al-Rahīm)