Al Haafidh Ibn Katheer said: 
Indeed it is mentioned in Saheeh Muslim from Abee Sa'eed al-Khudree in marfoo'  form, 'Whoever writes from me something other than the Qur`aan, then let him erase it.'
Ibnus Salaah said, 'And from what has been narrated to us about it is that it was disliked by: 'Umar, and Ibn Mas'ood, and Zayd Ibn Thaabit, and Abu Moosaa, and Abu Sa'eed in a group of late comers from the Companions, and the following generation.'
He also said, 'And from what has been reported to us regarding the permissibility of that or doing it: 'Alee, and his son al-Hasan, and Anas, and 'Abdullaah Ibn 'Umar Ibnul 'Aas in a group from amongst the Companions and the following generation.'
I say, that it has been established in the two Saheehs that the Messenger of Allaah (sallallaahu 'alayhi wa sallam) said, 'Write for Abee Shaah.' And indeed we clarified this topic in the introductions of our first books. And for Allaah is the praise.
Al Bayhaqee and Ibnus Salaah and more than one other than them have said: Perhaps the prohibition of that was for when it was feared that it would get mixed in with the Qur`aan, and then afterwards it was secured from that. And Allaah knows best.
Indeed it was related from the scholars of later times that they agreed upon the permissibility of writing the hadeeths, and this matter is exhaustive, widespread, and well known, without disapproval.
Shaykh Ahmad Shaakir said:
The older Companions differed about the writing of hadeeths. Some of them disliked it because of the hadeeth of Abee Sa'eed al-Khudree, that the Messenger of Allaah said, "Do not write from me anything except the Qur`aan, and whoever writes from me something other than the Qur`aan, then let him erase it." Related by Muslim in his Saheeh, and by Ibn Sa'ad (6/63) from 'Ubaydah as-Salmaanee from the older taabi'een, and verily the times of ignorance were realized after being written and erased. Most of the Companions were upon permitting the writing, and this is the correct statement. And indeed the scholars have answered the hadeeth of Abee Sa'eed with many answers. So some of them state that the hadeeth is mawqoof on him, and this is not good, for verily the hadeeth is authentic. And they have answered with other than this, saying that the prohibition was only for writing the hadeeths with the Qur`aan on one piece of paper, fearing that the two would get mixed up by one who was not knowledgeable about the beginning of Islaam.
And other late comers answered that the prohibition from that was particular to the one who was reliable in memorizing it. It was feared that he would rely on the writing, but if one is not reliable in memorization, then he should write. Indeed none of these answers are decisive.
The Correct Answer:
The prohibition was abrogated by other hadeeths showing it's permissibility.
So indeed Bukhaaree, and Muslim relate that circumstances required Abee Shaah to have something written from the Messenger of Allaah (sallallaahu 'alayhi wa sallam), which he heard in his sermon. This was in the year of the victory over Makkah. So he (sallallaahu 'alayhi wa sallam) said, "Write for Abee Shaah."
Abu Daawood, and al-Haakim and other than these two relate from 'Abdullaah Ibn 'Umar Ibnul 'Aas who said, "I said, 'O Messenger of Allaah (sallallaahu 'alayhi wa sallam), verily I heard something from you, so can I write it?' He said, 'Yes.' I said, 'In happiness and anger?' He said, 'Yes, for verily I do not speak in these two except truth.'"
Bukhaaree related from Abee Hurayrah that he said, "No one from among the Companions of the Messenger of Allaah Allaah (sallallaahu 'alayhi wa sallam) had more hadeeths than me, except for 'Abdullaah Ibn 'Umar. Verily he used to write, and I did not write."
Tirmidthee related from Abee Hurayrah that he said, "A man from the Ansaar was sitting with the Messenger of Allaah (sallallaahu 'alayhi wa sallam). So he heard a hadeeth from him, and it amazed him, but he could not memorize it. So he complained to the Messenger of Allaah (sallallaahu 'alayhi wa sallam). So he (sallallaahu 'alayhi wa sallam) said, 'Use your right hand.' And he signaled with his hand to writing."
These hadeeths, along with the affirmation of this action by the majority of the Companions, and the second generation, then the agreement of the Ummah after that upon it's permissibility - all of this proves that the hadeeth of Abee Sa'eed is abrogated, and that was the beginning of the matter when it was feared that it would distract them from the Qur`aan, and when it was feared that other than the Qur`aan would become mixed up with the Qur`aan. And in the Musnad (no. 7276) and in the Tabaqaat of Ibn Sa'ad (5/209) is what proves that a hadeeth was written for a lame man by them, and he read from it.
And the hadeeth of Abee Shaah came in the last stages of the life of the Prophet (sallallaahu 'alayhi wa sallam), and like that were the narrations of Abee Hurayrah. And it was in Islaam's later stages that 'Abdullaah Ibn 'Umar used to write, and he (Abu Hurayrah) did not write, proving that 'Abdullaah Ibn 'Umar used to write after Abee Hurayrah came to Islaam. If the hadeeth of Abee Sa'eed of prohibition was after these hadeeths of permission and permissibility, then that would have been known by the Companions with unadulterated faith. Instead, there came the decisive agreement of the whole Ummah after the conclusive proof that the permission of the matter was in later times. It is an agreement affirmed by many recurrent actions from every group from amongst the Ummah after the first period. May Allaah have mercy upon them all.
Indeed Ibnus Salaah said: "Then all that differing ceased, and all the Muslims agreed upon writing that being permissible, and were it not for writing them in the books for the lessons..." And indeed he spoke the truth. May Allaah have mercy upon him.
And from what proves that the writing of hadeeths has preceded from the older times of the Companions is what is found in Tahdheeb (1/470) in the biography of Basheer Ibn Nuhayk. Yahyaa Ibnul Qattaan said: "From Imraan Ibn Hadheer: From Abee Majliz: From Basheer Ibn Nuhayk who said: I came to Abee Hurayrah with my book which I used to write in. So I read to him from it and I said: I heard this from you. He said: Yes." And this text is also related by Tirmidthee (4/396), and by al-Khateeb in al-Kifaayah (no. 283), and Ibn Sa'ad (7/1/no. 162)
 The above is taken from al-Baa'ithul Hatheeth Sharh Ikhtisaar 'Uloomul Hadeeth p. 129-130.
 "marfoo'" This comes from the Arabic root word, 'rafa'a', which means: to be raised, or to raise. Marfoo' means something which is raised, in this case - a marfoo' narration is one that stops at a Companion in the chain of narration, but the text is such that none other than the Prophet (sallallaahu 'alayhi wa sallam) could have spoken it. It is used when the Companion does not state that he heard the narration from the Prophet (sallallaahu 'alayhi wa sallam), as is the case here.
 "taabi'een" This comes from the Arabic root word, 'taba'a' which means 'to follow.' The singular version of taabi'een is taabi'ee, which is a person from the second generation of Muslims.
 "mawqoof" This comes from the Arabic root 'waqafa' which means "suspended", technically - what is attributed to the Companions. It is also used for other than the Companions if restricted, e.g. mawqoof from az-Zuhree.
Some of the scholars have stated that this hadeeth by Abu Sa'eed al-Khudree was not a hadeeth at all, rather it was his own opinion. Shaykh Ahmad Shaakir counters this argument by saying that it is authentic, meaning that it is a statements which could not have been said by other than the Prophet (sallallaahu 'alayhi wa sallam) because it contains instruction about the preservation of Islaam.