All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Memorizing and reciting the Quran is of two kinds:
Firstly, it is an individual obligation to memorize the part of the Quran which one must recite in obligatory prayers, i.e. Chapter Al-Faatihah, without which the prayer is void.
Secondly, it is a communal obligation, if some people memorize the whole Quran the sin of the remaining people is waved out. However, memorizing the Quran is of great advantage and a good merit. The Prophet used to appoint Imams and leaders among his companions from those who memorize the Quran best. Abu Hurayrah narrated that the Prophet sent numerous groups and (before leaving) he told them to recite what they knew from the Quran. Then each one recited what he had learnt of the Quran until the Prophet came to a man who was among the youngest of them, he said to him: "What do you have O! (such man)?" He replied: "I have such and such and Al-Baqarah (Quran, chapter 2). The Prophet said: "Do you memorize Al-Baqarah?" He (the man) said: "Yes". He (the Prophet ) said: "Go, you are their leader, …etc." [At-Tirmithi] . In another Hadeeth the Prophet said: "People are led in prayer by the one amongst them who knows more Quran". All these are proofs that memorizing the Quran and taking care of it is of great importance.
As for the difference in reward between reciting the Quran from memory and reciting it from the Mushaf (the Arabic text of the Quran), Imaam An-Nawawi said the following in his book Al-Majmoo': "Reciting the Quran from the Mushaf is better rewarded than reciting it from memory because it combines reading and looking in the Mushaf which is yet another act of worship. This is what Qadi Hussain and others related. It is also the opinion of a group of the Salaf (rightly guided predecessors) and I did not come across any disagreement over this issue. Perhaps, what they meant are people who in both cases have the same level of devoutness and presence of mind. But the person whose devoutness and presence of mind increase when he recites from memory, then this is better for him." The writer of the book: "Mataalib Uli Ennuha" says: "It is a Sunnah to read the Quran from the Mushaf to occupy the sense of sight in worship."
Allaah Knows best.