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.