If we look at the Roman Empires "colonies" - it was typically to strengthen Roman power - a habit learned from their early days in Italy, and from the Greeks to an extent.
So if we assume, rather than like IOTL europeans, who colonised for prestige, religious freedom, and gold, the Roman Empire would have no motivation (at first). If there are religious conflicts, then that may have a factor - but the Romans? A reason I can see (If they discover the New World in the first place) is also gold. If they believe there is gold in Mexico for example, they may agree with the Spanish that the invasion is worthwhile, and then set up colonies to establish control.
The process would probably be able to repeat itself in Peru for silver.
Both circumstances would boost the Roman supply of currency, and then lead to vast gold to flow towards India.
Anything beyond gold/silver would probably be the result of a rise in popularity, or a need for security. Rise in popularity regarding tobacco? I can see the establishment of plantations after the fact. The same with cotton and sugar. Combine that with the Roman habit of wanting to produce goods like silk domestically, then I can see that and other eastern goods being stolen and grown in the new World over time. (The idea of Spice plantations in the Yucatan is a fun image - combined with the Amazon rainforest).
But I can't see them having the idea of Manifest Destiny, or feeling the need to conquer the Great Plains - taking Mexico, Peru and the Atlantic Seaboard is probably the most likely, with any further activity being trade/fortification. (Admittedly, I have no idea what Brazil would be like ITTL). There isn't any obvious benefit to them beyond.
Something interesting, is that in my estimation, the establishment of colonies would probably leave the Roman Empire reducing trade with India excluding luxuries like spices, making the initial colonization for gold less important.
I don't think a Roman Colonial Empire would be as vulnerable to independence as the other powers - a larger home manpower base, and fewer European threats to be worried about, and a history of resettling restive populations, would probably mean that huge amounts of people from the Americas (Primarily Mexico/Peru) would be forcibly resettled in Europe.
TL;DR Resource driven, control colonies, vast forcible population resettlement if there are invasions - and a reduction if not end of the flow of gold out east as the West produces the resources typically grown out east.