I just (3 weeks ago) bought an '06 Heritage Softail Classic, so I can comment fairly on the contrast between it and my '05 America.
First, I think that Vegas' comments above are a bit outdated. Not saying that that sort of thing doesn't still go on, but Harley dealers have, more or less, returned to Earth. I got my FLSTC for $500 under MSRP, for example, and it had NO dealer add-ons like a chrome front end, etc.
Riding it is, well, different. Where the America is lithe, light and flickable through curves, the Harley conducts itself with careful dignity. It gets there, but in its own time and in its own way. You either love it or can't deal with it.
Pretty much every important stock component is of very high quality, whereas the reverse is true with the America, where cost-saving is evident in every important component, such as the horrible shocks, for example.
On the highway, the Harley rolls down the road as if you were riding a sofa, soaking up almost every wind gust and expansion joint as though it wasn't there. Can't exactly say that for the America. The same weight that is such an annoyance in parking maneuvers becomes a huge asset on the road.
Functionally, the America is a terrific bike, where the Harley is perhaps less terrific. That said, the Harley engine, like the Triumph's, is pretty much bullet-proof. Like the Triumph, to maximize power you need to change out the exhausts and either re-jet your carbs or re-map your fuel injection. In neither case is that a big deal, and in both cases it ads a significant amount of torque and a bit extra horsepower. Harleys, of course, being all about torque, and much less about horsepower.
Finally, the Harley (some models, anyway) is something the Triumph isn't and can never be: a rolling, evocative, work of art.
That said, I will add that "art" is in the eye of the beholder.
Finally, I think that it's best to take Harleys (or not) on their own terms, and not compare them to any other bike. To do so is irrelevant and, at the end of the day, silly.
Just my opinion; I could be wrong.
Here's the H-D the day I picked it up (12/31):
