well if you didn't really have any problems before with what you were doing such as lag or stuttering, then you shouldn't with a processor without dual core.
There are some programs that will run better on a dual core system like some photoshop or 3d programs that can utilize it.
The other reason you'd want dual core is for like burning a cd and playing a game or surfing the internet at the same time.
If all your doing is surfing and listening to music, then you probably don't need a dual core processor.
maybe this articles will help you:
http://www.computerworld.com/blogs/node/2070
http://www.devhardware.com/c/a/Compu...r-Double-Hype/