Assalaamu alaykum. I read that Prophet Muhammad, sallallaahu 'alayhi wa sallam, did not excute the prisoners of war and that Allaah became angry with him. This was when the Prophet and Abu bakr were crying. I am a bit confused; was this not merely a case of wrong itjihaad (independent reasoning), in which case one would be rewarded for a wrong decision and rewarded double for the right one?

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.

The story of the prisoners of the battle of Badr to which you referred in the question is mentioned in a long hadeeth that was reported by Muslim and others and narrated by Ibn ‘Abbaas from ‘Umar ibn al-Khattaab .

Al-Qurtubi stated that most interpreters of the Quran are of the view that the blame was directed to some Companions, as he said in his Tafseer: 

‘‘This verse [i.e. verse 67 from Chapter 8] was revealed on the day of the Battle of Badr as a rebuke from Allaah to some Companions of the Prophet .

The meaning is: you should not have done this act [suggesting to take a ransom from the prisoners instead of killing them] which led the Prophet to take prisoners ... Allaah says: {You desire the commodities of this world}. The Prophet did not order to keep the men at time of war, and never did he want to seek the pleasure of this world; but it was the majority of those who took part in the war who did it. The blame and rebuke were directed to those who advised the Prophet to take a ransom. This is the view of most interpreters of the Quran, and there is no other correct opinion.

The Prophet was mentioned in the verse because he did not object to their suggestion when he saw this after the battle of Badr; meanwhile, Sa’d ibn Mu’aath, ‘Umar ibn al-Khattaab and ‘Abdullaah bin Rawaahah disliked that. The Prophet was preoccupied by the greatness of the matter and the descent of victory, so he did not object to keeping the prisoners (and accepting the ransom). It is for this reason that Abu Bakr and the Prophet cried when the verses were revealed.”

Thus it becomes evident that the verse was addressed to some Companions and that the blame was addressed to them due to them favoring the worldly life.

Allaah knows best.