[1:1] i(l)-RaHman i(l)-RaHeem

 ...i(l)-RaHman i(l)-RaHeem [1:1]
(the Merciful the Merciful) - two beautiful names of God

Both derive from the same root: "ra, Ha, mim" R-H-M) - meaning whomb or (R,H,M,M) Mercy ("raHma")

So what do the different forms mean
Ra-Hee-M
Ra-H-Man

The names are similar in meaning ("Merciful") and many scholars have attempted to explain the difference:
e.g.

• It indicates a different meaning ("Merciful to all, Merciful to some") - although there is no proof that this was the view held in the time of the Prophet (s.a.a.s.).
• It is the superlative form (i.e. "Most" Merciful) - but this is not true since the superlative of the root R-H-M is "aR-Ha-M" (More Merciful) or "al-aRHaM" (Most Merciful)
• One is a pure Arabic word (Raheem), the other predates Arabic (RaHMan) - this is my view

The arabs at the time of the Prophet Muhammad (s.a.a.s) did not actually know what the word "RaHman" meant.  It was foreign to them. 

Suhail said: "As for 'RaHman', by God I do not know what it means"

 In fact the name RaHMan predates Arabic

• it was known and used by non-Arabic speakers - i.e. the prophet Abraham (a.s.) [Quran 19:45] and the Virgin Maryam (a.s.) [Quran 19:18].
• it is still a name of God in Hebrew "RaCHMan" (R-H-M) meaning "Merciful" - evident in some jewish names like "Rachmaninov"