Difference between intel core i3 processor & core 2 duo

i3 is 32 nm , core 2 duo is 45nm , i3 is more economic , less heat , and is faster.
i3 and core 2 duo are both dual core.
Quad core have 4 cores , the diference between old quad core and i5 is the same like the dual cores.
If you use software what is CPU dependant you need an quad core ( i5 ). Slowly all the games, software and aplications are made to use multicore ,then a quad core is the best choice and is more future proof.
For a budget PC used for gaming an i3 is enough but you don't know for how long.

If you have a 2nd gen Core i3 (Sandy Bridge) and a Core 2 Duo clocked at the same speed. the Core i3 will be on average 20% more powerful.

The Core i3 has hyper threading, Core 2 Duo does not.

Core i3 also has the GPU on the same chip, the Core 2 Duo is just a CPU