Kill versus capture
If they had captured Osama bin Laden, what would they have done with him? Barack Obama opposed the Guantanamo Bay facility, so they would have had to keep him on United States soil. Where?
If captured, he would have been tried, convicted, and executed. So not kill versus capture but kill then versus later.
His trial would have been an obvious target and excuse for retaliation.
His trial would have kept him in the news, encouraging more attacks.
He was perceived as dangerous at the moment that he was shot. Source
When the commandos reach Bin Laden’s room on the third floor, an AK-47 and a Makarov pistol are seen in arm’s reach of Bin Laden.
While there have been times and circumstances when the US military has taken prisoners intentionally, that's not really their job.
Burial at sea
If he had been buried on land, his grave would be a landmark. Supporters might use it as a rallying point or symbol. Opponents might vandalize it.
No grave makes it harder to focus on a location.
Obama's words in a 60 Minutes interview:
"It was a joint decision. We thought it was important to think through ahead of time how we would dispose of the body if he were killed in the compound," says the president.
Note that this also indicates that it was not guaranteed that bin Laden be killed at the compound. Capture was on the table.
- Ick. Who wants to see that?
- Another thing to make people mad.
Obama said in the interview:
It is important for us to make sure that very graphic photos of somebody who was shot in the head are not floating around as an incitement to additional violence. As a propaganda tool. You know, that's not who we are. You know, we don't trot out this stuff as trophies. You know, the fact of the matter is this was somebody who was deserving of the justice that he received. And I think-- Americans and people around the world are glad that he's gone. But we don't need to spike the football. And I think that given the graphic nature of these photos, it would create some national security risk.
Apparently they felt that the risks of a capture were worse than the risks of a death. So the mission parameters were such that they shot him rather than attempt a capture. Although apparently there were some circumstances in which he would have been captured, the reaction to the actual circumstances erred on the side of safety for the raiding party.
Once he was dead, they seem to have focused on not making him any more of a martyr than he already was. That was more important than proving his death.
Obama, from the same interview:
You know, the truth is that and we -- we're monitoring worldwide reaction. There's no doubt that Bin Laden is dead. Certainly there's no doubt among al Qaeda members that he is dead. And so we don't think that a photograph in and of itself is going to make any difference. There are going be some folks who deny it. The fact of the matter is, you will not see bin Laden walking on this Earth again.