Knight wins. He has superior armor. He has a weapon and training optimized for individual combat.
The Roman is very, very well trained. But it's as part of a formation. He arms and armor are built for that as well. Even if they're both on foot, the Roman loses.
In a mass combat, the outcome is less clear. A medieval army had very strong cavalry but very weak infantry. The Romans weren't used to cavalry like that, but I'm not sure they couldn't have handled it. Certainly they would shatter the medieval infantry in short order. Historically, knights had trouble against discipled ranks of well-trained and well-equipped infantry. The Roman obsession with march security and well-defended encampments make a cavalry raid strategy infeasible.
Medieval armies' structure also made them less responsive to good generalship than the well organized Roman legionary organization. Feudal European armies mostly fought against similar opponents. Whereas the Romans routinely fought all kinds of different enemies. Overall, I'd give the edge to Rome in a mass combat.