I doubt, to the point of virtual certainty, that the $20bn bought immunity from prosecution.
You asked whether there "some other kind of damages they might've had". They didn't pledge the $20bn to avoid other damages. It's basically protection money, to get Obama off their backs.
As I've pointed out elsewhere, the rhetoric being spouted by Obama was seriously affecting the market's confidence in BP and that was making it difficult for them to raise short-term funding (which is how a company of BP's size operates on a day-to-day basis). There was a real risk that BP could run out of cash - i.e. go bust, essentially.
Now, you expect that sort of lunatic, xenophobic rhetoric from Congress and, even if a couple of representatives from Bumfuck, AZ and Shitsville, TN decide to introduce a bill that would seize all of BP's assets, the markets aren't going to take it seriously...unless it looks like the President might actually sign it into law. The fact that Obama and the White House were demonising the company and blatantly stirring up public opinion by constantly referring to it as "British" Petroleum (it's not been called that since the merger with Amoco over a decade ago) made them nervous.
From the BP board's perspective, there were two issues at stake. Firstly, straightforward survival. Obama had clearly identified BP as a scapegoat. He clearly didn't care whether BP was to blame for the oil spill or not - he needed someone to blame and BP were the prime candidate. Arthur Andersen's experience being scapegoated for the Enron collapse showed that the US government is prepared to effectively lynch a company, even if they're innocent (Arthur Andersen was eventually exonerated by the Supreme Court, by which time it was far, far too late to save the company). BP needed to avoid that at all costs.
The second issue was that BP needed to ensure that it could continue to do business in the US. BP has a huge footprint in the US - more than 40% of BP is what used to be Amoco. If BP didn't placate Obama and get him back on-side, they may well have faced legislation targeting them in the US. That would certainly have had a seriously detrimental impact on their future profitability and may have forced them to sell their US operations at a knockdown price. BP's board probably felt that $20bn over four years was a price worth paying to avoid that.
Ultimately, they went into their meeting with Obama (at which the $20bn was agreed) with a carrot and a stick. The carrot was the $20bn - basically, they agreed to pay the protection money. Sure, no one like to be the victim of a shakedown but shit happens.
The stick was the risk of BP going under. If that happened, the entire cost of the clean-up effort and compensation would have had to be borne by the US government and Obama would have been crucified had it emerged (and the BP board would have made sure it did - in fact, they would have completely trashed him in the press) that he had turned down a $20bn peace offering from a company that was only liable for $75m.
So, end result is that BP eats humble pie and coughs up 18 months' profits spread over four years (and there's a chance that the total compensation bill won't even come to that much), while Obama walked out to tell the press corps that "BP is a strong and viable company and it is in all of our interests that it remain so."
A minor side-effect is that the United States' attractiveness as a place to do business has taken a knock. Multinational corps will be just a little bit more wary of doing business in the United States from now on. If the equivalent of a BP-Amoco merger were to occur today, there's a good chance that the Amoco name would be retained and that it would be given a separate stock market listing, to insulate the parent company from the US' banana republic-style political foaming-at-the-mouth lunacy.
Another impact is that Obama has damaged US-British relations. Remember that, whether you support the wars in Afghanistan and Iraq or not, Britain has been the United States' primary ally in the war on terrorand British troops continue to fight against the Taliban in southern Afghanistan and there are regular deaths (most recently yesterday). This, despite the fact that, in standing alongside the US, Britain has put herself at greater risk of terrorist attack.
Obama's xenophobic approach was a massive slap in the face. You simply do not do that to your closest ally.
So, while it might look like Obama got a good result, the whole thing was, in fact, an unmitigated fuck-up on his part.
