The primary differences between the Core 2 Duo and Core i3 is that the Core i3 has:
1. An integrated graphics core (model # depends on the generation / model #); but 1st generation Core i3 had no integrated graphics core.
2. Support for DDR3 RAM
3. Lower power consumption
4. Higher performance.
In terms of RAW CPU performance increase from Core 2 Duo to Core i3:
C2D -> 1st gen Core i3 = 10%
1st gen Core i3 -> 2nd gen Core i3 (Sandy Bridge) = 12%
2nd gen Core i3 -> 3rd gen Core i3 (Ivy Bridge) = 6%
3rd gen Core i3 -> 4th gen Core i3 (Haswell Bridge) = 6%
Assuming you are going from a C2D to a 4th gen Core i3 with the same clockspeed, the Core i3 would be about 38% more powerful.
Core 2 duo was the brand name for intels dual core processors produced from 2006 to roughly 2009. When intel moved the memory controller from the motherboard to the CPU die, they changed the naming scheme. The dual cores are now called core I3.
Low end dual cores since the core 2 duo's release have been branded as pentium and there were a few early core I5 parts in 2010 that were dual core.