Core I3 are ussully 2 cores vs Core 2 Quads are 4 cores, now due that i3 are newer technology which makes them faster they probably are equaly capable performing CPU demanding tasks. If you already own a quad processor, youll gain little to nothing by swaping, but if you dont better get an i5 they come with 3 and 4 cores and are alot faster than a core 2 quad.
Some of the Core 2 Quads are better, but also more expensive and will likely force you to a socket 775 board that will require DDR2 which is also more expensive. The i3 is a much better choice unless you already have a socket 775 system and are upgrading it.
Sorry I forgot to tell you about the PC for under 500: If you already a Core 2 quad pc, all you need to do its get a video card, something like a GTX460 will do, theyre about $150 and a 24" LCD hdmi monitor for another $150 thats $300 dollars to bring your pc up to gaming specs.
Now if you want to go all out with an i7 hexacore processor, you're going to need well over a 1000 bucks. There is just no way to get a decent gaming PC for under $500 even if you build it yourself. I just showed you that the least expensive graphics card and monitor are around 300.
The only other way is to buy a used on on craigs list or ebay, but you still will need the HD monitor.
Anyways good luck and keep on searching.