I would argue they were treated much more similarly than many would think, with Roman slavery being horrible and somewhat comparable to the Atlantic Slave Trade (even if I think its healthy to be more uncomfortable with the race based slavery that has an ongoing legacy)
This is a bit of a pet issue of mine since I feel that there are quite a few reasons why the horrors of ancient slavery (in particular Greece/Rome) are ignored. I (or we) don't know enough to entirely answer the question but I do think its important to inform that we do know that Roman slavery was pretty nasty.
The overwhelming majority of Roman slaves were not household or city slaves but field slaves and other labour intense occupations who worked in horrible conditions. Mining in particular was a death sentence, if I recall correctly from Carthage Must Be Destroyed for a 500+ year stretch at any one point there were over 40,000 mining slaves in Spain alone. That's with a mining life expectancy of 3-5 years. If you read some of the descriptions of Roman field slaves it is reminiscent of the conditions of plantation slaves in the America's.
There was also the lack of rights for slaves throughout much of Roman history. Owners could essentially do whatever they wanted to their slaves, sex could not be refused and it was a fact of life that slaves could be raped. There is a range of sentimentality of these house slaves that we see on things like grave stones but again these slaves are a small minority of slaves.
Now a lot of this (the horrible conditions, rape, ect) is similar to the slavery in the America's but without the racial/religious aspects of the Atlantic slave trade (at least in America, the racial aspect gets more complicated in other places). I don't think we have the hard numbers to properly compare and contrast to my satisfaction but my perception of the issue is that they really are similar. People generally don't think of mining slaves when they think of black slaves but Brazil in particular had hundreds of thousands of slaves die in the mines with an even lower life expectancy than the Roman slaves in Spain. I do think that this is an example of one trend which is that as horrible as Roman slavery was from what I've read the comparable hard labour job would often be even worse in the America's (because of things like tropical diseases/conditions, racism, profit margins, ect) .
Tldr: Surprisingly similar in many horrible ways but on average slaves in the America's would be treated worse.
Of course there should also be the usual disclaimer about Rome being a very long lived empire so even "ancient Rome" arguably spans over a 1000 years and especially post Christianity there are some changes.
Sources: Carthage must be Destroyed, Slavery in Brazil (not the best source), and Roman Slavery