As an outsider, the political problem with Obamacare is obvious - it simply overpromised massively. Not surprisingly, it turns out that many people will have to pay a bit (and in a large number of cases a lot) more so that other people can pay a lot less and/or get a lot more out of the system - that's just how the maths works out. So Obamacare is, and always was, fundamentally a political deal between generations and between classes about who pays for what - but that was never made clear enough at the time, in my opinion because the Obama administration was far too worried about its short-term poll ratings and was (from what I could tell at the time) obsessed with the law being seen to be "popular" at the time it was passed.
I suspect this is where Romney's and Obama's proposals differed most - in the rhetroric which would have been attached to a Romney-led reform. Because Romney would have been coming from a position of political strength (Democrats would have been in favour of any kind of reform in the direction he proposed), he could have afforded to make the "grand argument" without taking much risk; whereas for Obama the risks were greater, hence he did not.
The problem the administration now have is that because noone was straight about how this would work at the time, a lot of people feel that what is now happening was never the political bargain that was entered in to. As a consequence, they feel that they and their fellow voters (including, crucially, those who were in favour at the time) were taken advantage of. Many of this group would likely have grudgingly supported the actual deal as a case where effectively their "side" was outvoted if it was concluded more openly at the time - but the way the changes were marketed leaves them feeling like the administration lied to people to get the law passed.
This leaves the administration with a problem which cannot be solved in any satisfying way without simply repealing Obamacare and starting again after a fresh election with a fresh president. As this is unlikely to happen I suspect the programme will be unpopular for many years to come and get blamed for all sorts of ills, some of which will be its fault but many of which will not.