The Core2Duos are several generations older, and slower.
As far as the i3 being good for gaming, depends on what games you're playing and at what resolutions, video cards are the most crucial factor when choosing a system to game on, but you should never go too cheap on the CPU either.
i3s are hyperthreaded dual cores, they have a newer and better architectures than the C2D, C2D has more L2 Cache, i3 has more L3 cache, i3 is part of the LGA 1366 and LGA1155 sockets and C2D is LGA 775, the GPU is in the same chip as the i3 CPU, i3s don't have FSB, uses DMI instead, i3 has newer stepping instructions.
Generally speaking, rough average performance increase over Core 2 Duo / Quad; assuming same clockspeeds:
1st Gen Core i3/i5/i7 = 10% increase
2nd Gen Core i3/i5/i7 (Sandy Bridge) = 23% increase
3rd Gen Core i3/i5/i7 (Ivy Bridge) = 29% increase
Current AMD Phenom II and FX CPUs are basically comparable with Intel's Core 2 Duo / Quad CPUs in overall performance. However, there are specific type of benchmarks where the FX does perform better than the Core 2 Duo / Quads, such as video encoding and file compression.
The move from the Intel® Core™ 2 Dual to Intel Core i3's was the moving of memory controller from the board to the processor. Also with the Intel Core i3 we would add hyper-threading and an IGP (Integrated Graphics on Processors). As Jag pointed out these changes would yeild almost a 30% performance improvement at the same clock speed.