The job of a shock absorber is to keep your wheel in contact with the road at all times, even if the road gets bumpy. The better it can do that, the better your ride will feel and the better it will handle (losing traction from a wheel bumping changes how well your bike sticks to the road).
A bad shock absorber won't fully absorb the shock of a wheel hitting a bump and will transmit that up to the bike and the rider. This will be felt like jolts and harsh, well, shocks that get transmitted to the rider. It will also not keep the wheel fully against the road.
There's a whole science behind it but there are basically two components involved, the spring and the damper. The spring is what flexes to extend and contract the shock absorber, and the damping system is there to keep it from just pogoing up and down uncontrollably. Both matter.
So yes, aftermarket shocks will make the ride more comfortable, keep the bike more stable and controllable in turns and in general improve handling and how fun the bike is to ride.
The America seems to have the same KYB stuff that was on my Rocket Touring, at first glance... not that great. The Bonneville lineup standard shocks are also harsh. I went with a set of Progressive Suspension 444 HD on the rear, made a huge difference. Small washboard bumps that rattled my fillings just aren't felt at all anymore, and the bike tracks in corners as if on rails. Under $600 a pair at places like JC Motors or Amazon (or both) which is pretty decent for high quality shocks.
For the front, you could get Progressive's springs also, though another interesting approach would be Ricor. They make a product they call the "Intinimator" (no, not a misspelling) that is very clever. It basically completely changes how the front fork shock absorber works and only costs under $200 at the moment. I got a set for my bike and will be trying them this season, so far I have no experience but all reviews are glowing.
They do rear shocks, also, and assuming they work the same they are probably also of interest.
TL/DR: Shocks matter. :p