The main difference is that the Core2 Quad are a Quad Core CPU whereas the i3 are only 2 core. Therefore the C2Q will out-perform the i3 in most multi-threaded situations. However, clock for clock I would say the i3 is faster since it is a newer chip and obviously the technology is moving forward.
My advice would be if you are thinking of upgrading and already have an LGA775 board then go for the Core2 Quad. However, the LGA775 platform is effectively dead now, meaning that no new chips will be made for that socket. If you are starting a new build or are upgrading your motherboard anyway then I would go for the i3 or even the i5 or the i7. Certain models of i5 are quad core processors and perform very well in gaming and every other task. If you have a larger budget maybe the i7 is for you Hope this helps!
Core 2 Quad is a true 4 core CPU, the Core i3 is a dual core processor that is hyper-threaded, so it can run 4 threads at a time. In my opinion, the price of the i3's makes them pretty much worthless because you can get a true quad core CPU for much less from AMD. If you need a budget CPU to upgrade a socket 775, get the C2Q, but if you are building a new system, get an i5 750 quad core if your budget allows, but if you can't spend that much, get an AMD quad core for much less (2.8-2.9 GHz AthlonII x4's are only $100-$110!).