The Mongols trashed the Persians and a large portion of the Middle East so badly during their expansion that they utterly destroyed centuries of infrastructure setting back the region quite a bit in development. So the Persian Empire wouldn't be an issue.
As for the Romans, they tended to have difficulties when facing more mobile forces that used a lot of calvary and skirmish tactics like the Parthians for example. The Mongols wouldn't have fought the Romans "on their terms" they would've used hit and run tactics and drew the Romans into a fight that would've been more advantageous to them.
The problem with the Mongol Empire was that it gradually lost its "identity" soon after Genghis died. By the time Genghis' grandson Kubla came around, he was pretty much living like a Chinese noble rather than a nomadic horse warrior. Basically they suffered the same fate most barbarian empires do, they got "civilized" and embraced the cultures they conquered.
All that being said, I'd have to say that the British Empire had a longer lasting influence on the world today. There's a list of reasons, but one of the more obvious ones is the fact that lot of people use the English language to communicate throughout the world. Even if it isn't someone's first language it tends to be taught a lot for business and trade purposes.
The US, Canada, and Australia are basically its "children."