The great thing about dual core will mean an overall smoother experience for the user. As of right now your computer has different processes that are run to according to what your OS sends it, you can see this if you alt ctrl del and look at your processes in Win2k/XP. In reality your processor is doing a small bit of executing code for each of those processes that request it, then switches to the next and so on. With dual core processors that work can be split up even further. So basically your gonna be able to do things like play a game , and burn a dvd at the same time , multi tasking is where the user will see the biggest most obvious increase of performance. Later on when programs are designed from the ground up to take advantage of two cores a single app, like games will also see great gains.
The server market is where dual core will make a big big splash, with boards that can handle 2 - 8 cpus, you will effectively double how many processors your have to handle the workload how awesome is that!
As far as amd vs intel, its still up in the air, some people think that the A64 is going to have an advantage with its HT link, and the fact that the K8 was built from the ground up to eventually go dual core. On the flip side the Intel dual cpu will have a shared bus, or possibly 2 buses. Its kinda irrelavant though, because intels successor to the P4 is coming which is probably why intel is putting more resources behind the EE dual cores with HT, not only can they charge more and claim more features for their top of the line cpu, they will have a succesor ready anyway when dual core starts to replace single core in the fabs.