Hillary is less likely to depart from Iraq in the same way, but overall, she'd have a slightly more hawkish version of Obama's pragmatic foreign policy. She would also surprise many by being more liberal than her last name would suggest; she wouldn't pursue the ACA, for instance, though some form of Dodd-Frank is inevitable.
The economic recovery won't be any better than OTL. I think the reason the recovery hasn't spread evenly is not because of Democratic policies, but because of a persistent fiscally conservative strain. As a matter of fact, it has happened with every recovery of every recession in the last thirty years. Though Obama, and likely Hillary as well, would be more progressive than any POTUS in the last fifty years, that's not saying much. Unless Hillary beings about a progressive revolution (unlikely - the nature of politics don't fulfill it, and like Obama, she's a pragmatic and compromising progressive), likely we see another flawed recovery (and worse without the ACA as because of it, lowered per capita healthcare costs meant more money could be spent).
In regards to LGBT rights, the last eight years have seen a revolution in that regard. Hillary revealed her progressive views in that regard to great controversy (as usual - everything she says is seen as controversial!) in 2011. She'd likely do it earlier than OTL. Republicans would denounce her "far-leftism" and reuse their bag of sexist insults from the 90s. The backlash from her pro-LGBT comments (because 2010 was a fair bit more conservative than 2016) may be what costs her the House in the midterms.
In 2012, her victory is pretty much inevitable no matter who she faces. She's still probably killed Osama bin Laden, and she's likely to campaign as a technocrat and a policy wonk. She's not charismatic, but she has an air of "knowledgeability" and that she is a ready hand. So, she likely wins in 2012, more than anything because the Republicans have a bad field of candidates (if they choose Romney, he'll be seen as an elitist billionaire; if they choose Santorum, he'll be seen as a reactionary).
So yeah. TL; DR Hillary would be slightly more hawkish and more pro-LGBT rights, but otherwise not very different from Obama. For all that 2008 was touted as a battle of the ideologies, it was little more than a popularity contest.