In some games (but not all games) 4 cores is faster than 2 cores.
Other programs like video rendering can see big improvements with 4 cores
Core 2 is actually an architecture like Nehalem.
If you are asking what the difference between a core 2 duo and a dual core is, its the same as the difference between a Kleenex and a tissue. Core 2 Duos are dual cores, but there are many other dual cores like the Athlon x2, Athlon II x2, Phenom II x2, i3, and Pentium D.
The core 2 quad is essentially 2 core 2 duos on the same die. If you use applications that can support multi-threading the core 2 quad will show a HUGE performance improvement over the core 2 duo at the same clock speed. If however, you don't use heavily threaded apps and just do things like gaming, the Core 2 Duo 3GHz or higher or the Phenom II X3 / Athlon II X3 are the best choices because buying a Quad-Core CPU for things that don't use 4 cores is a waste.