Do 20-bore guns recoil less than 12-bores?

Q: Can my 20 have the same power as my 12-bore shotgun?

Providing the cartridges have the same muzzle velocity, then striking energy and range will be the same.
And, providing the 20-bore and the 12 are choked similarly, the patterns should not be much different either.

However, if you are going to shoot 28g through your 20, expect the recoil to be higher. Gun weight has a lot of influence on the recoil you feel, so with your lighter 20 you might wish to drop down to 24g loads.