While I agree in part with the sentiment, I think is totally unfair to consider ancient slavery in Greece or Rome as less cruel. It was not less cruel depending on the slave in question. Slaves in mines and agricultural estates were in worse conditions then anything in American south. But if you were an educated slave then your life was indeed better. That also means that was common for slaves in ancient Rome to be able to buy freedom. Slavery was everywhere in society, so the comparison is really hard to make.

There is indeed a racial component in colonial slavery that was not present in ancient Roman slavery. A slave could be from Germany or from Syria and there was no difference in treatment.

I would say that both late trans-atlantic slavery and nazism share a philosophical root in the eugenetic movement, but both grew in parallel with different motives: in one case a justification for economic exploitation, in the other an ideological tool to enforce unity in nationalism.

The transatlantic slave trade started before the concept of race and the eugenetic movement. During the 15th century the justification was more routed in religion and the idea of having prisoner of war being better then to kill the enemy. Still and excuse for economic exploitation, but maybe more akin to what the greeks and romans were doing.