Both ahadith say that writing down hadith was or is not allowed!
The first hadith is also emphasizing on only telling the truth about Muhammad and warning those who attributed any falsehood! In fact allowing to tell or to narrate hadith. So an oral transmission is allowed and this goes along with QuranQur'an. As it clearly orders the mothers of believers to tell or transmit what they hear in their houses, no matter if it is QuranQur'an or hadith:
I'd like to point at the fact that even if QuranQur'an was already written on pieces of what ever came in hand it was not collected as a book at the time of our Prophet and compiling it in a book was not an easy matter, as people were aware that what they might be doing is something the Prophet himself never did, so it needed a discussions and argumentations to realize it!
Some explain those hadiths as follows: they should make a clear cut between revelation and hadith. So one could understand it as when there's a revelation don't write down anything else! As there are ahadith indicating that some of the scribes were listening and writing not only on times of a new revelation. So it is narrated that in Ubay ibn Ka'abs private mushaf he added what some may know as du'a al-Qunoot "Allahuma inna nast'eenu bika wa nastaghfiruka ..." as a surah. And mixing Quran the words of Allah with other things or hiding parts of it or adding to it is a serious matter as we know from Quran like one can read in verses like 2:42, 2:74,2:79 and 3:187.
the hadith addresses people who were close to idol worship. They beside of this were (to some extent: 'Omar was not allowed to copy and keep copies of the thorah, while Zaid ibn Thabet even was asked to learn Hebrew to be able to read it) frowned upon to read in the books of people of the book or copy them.
Also it might have been a necessary command as the QuranQur'an at the time was not yet complete and maybe some verses were abrogated and if we think of the later collection of the moshaf, this was a big help for Zaid ibn Thabet and others!
Beside the confusion issue the words of Allah should be treated with as respectful as possible, so they shouldn't get mixed with the words of a human and the Prophet has chosen to have them written down to make them fix. While the words or doings of the Prophet are witnessed and have a lower value, so writing them down would make the word of Allah equal to the word of the Prophet and again as they were close to Idolatry this would make the matter risky! Malik narrated a statement of 'Omar -who at first wanted to compile a hadith book- saying: "There shouldn't be a book beside the book of Allah!"
Ibn Hajar also added that this prohibition was temporary: that means it was prohibited to the scribes to write while they were listening to a new revelation. (I'd like to know whether he counts abu Sa'id al-Khudri among the revelation scribes?)
Also note the fact that possibilities to write were not given (they didn't have paper or enough material) so they used bones, pieces of wood or palmed stalks or thin white stones ... in short what ever could be used to be written on. If they did the same for hadith they might have a problem!
I'd like also to point at a matter I've experienced: It is hard to memorize two things at once. And it helps a lot to write things down to memorize them and to read them from your own transcription. I've even been told by a hadfidh of Quran if one wants to memorize it based on a moshaf it is necessary to always use the same printed edition (or at best always the same copy).