From what I've read elsewhere and from the Hierarchy Charts here, it seems to be pretty much on par with the 8600GTS I have at the moment
Both are PCI Express. The 8600GTS has 256MB of memory, a huge fan and requires additional power.
The 6450 on the other hand has 1gb of memory, a passive heatsink and no extra power requirement. It also has the bonus of supporting DirectX11.
Now I feel as if I should switch to the 6450 but my last experience with a Radeon was many years ago with the 9250 and that was a nightmare when trying to get Knights of the Old Republic to work. So I've been a bit biased against them ever since.
So would it be worth me switching to the 6450? Are they easier cards to work with now or should I stick with the 8600GTS?
It would absolutely be worth it. AMD has come a long way, and driver problems and such are no more of a problem for AMD cards than Nvidia. As you said, your last experience with AMD was many years ago--it's not many years ago any longer. It's now. You should switch to the 6450 if only for the fact that the hardware is newer, cooler, and runs more efficiently. The gaming performance won't be super spectacular, but it certainly won't be a step down from what you have now.