The equations are a bit more complex that this, because it depends on the wire run length.
However, 10 gauge will have less electric loss than 12 gauge, and for anything but a short cable run, it may start to heat up at 240v-20a (240v-16A continuous) whereas it wouldn't at 120v-20a.
You're wrong. There is no difference in heat generated from a 120V-20A outlet that pulls 16A continuously vs. a 240V-20A outlet that pulls 16A continuously. Both wires (hot and neutral, or hot1 and hot2) are moving 16 amps of current continuously in either scenario, so the losses from heat are identical.