The difference is that the existence of birthers is actually a pretty good boost to the Democratic base. "Look at these crazy Republicans, they're so nuts they think he's not an American citizen! Donate to help us fight the hordes of ignorance!".
This comes off a lot more like "People have got to know whether or not their president is a crook. Well, I'm not a crook." And then cue the missing evidence. Nixon was sunk by less than twenty minutes of audio. Two years of email?
After all, in Washington they're not bothered if you're corrupt. They're bothered if you're corrupt and can't hide it.