The American Scene's James Poulous lays out what I think is the key distinction:
Atop other impressive one-liners, Gerson produces this classic: “Perhaps Obama is just conventionally liberal.” Well, duh. It’s stunning to see Gerson, of all people, confuse post-partisan politics with post-ideological politics, but he does it consistently with Obama. Obama never vowed to transcend Liberalism. He repeatedly vows instead to abandon permanently mobilized, rigid, say-anything, do-anything party warfare. It’s fanciful to insinuate as Gerson does that Obama voted against John Roberts because he’s a partisan hack.
Liberals, too, continue to misunderstand this, and assume that whenever Obama talks about moving past partisan politics that he's just going to give everything away to the Democrats. See, for example, Paul Krugman:
My biggest concern about an Obama administration is that, in the end, he won’t make universal health care a priority. My second biggest concern is that “Unity” means never having to say you’re sorry: that in the name of putting past partisanship behind us, the next administration will sweep the abuses of the past 8 years under the rug, the same way Bill Clinton did in 1993; the result of that decision was that the very same people responsible for Iran-Contra showed up subverting our democracy all over again.
If you want to know the real difference between Bill Clinton and Barack Obama, this is it. Clinton was a partisan moderate -- he fought like hell against the Republicans, but also co-opted their ideas and moved his own party towards the center in order to gain the White House and later, to stop the bleeding. But in the process he left a mixed legacy with both blue-collar Democrats (free trade, welfare reform, etc.) and cultural progressives (the Defense of Marriage Act, Don't Ask Don't Tell, etc.).
Obama, on the other hand, could be called a post-partisan liberal. This doesn't mean he isn't a political animal -- he absolutely is. But ideologically, he is much farther left than Bill Clinton, and probably Hillary too.