Presumably you'd have some sort of technological exchange. Looking at
a relevant Wiki article, the most likely exchanges are paper and decent steel from the Han to Rome, and better engineering tech both ways.
It's hard to imagine a whole lot coming out of it, I think. Arguably, Rome's decline started in the late Republic; Roman cities were drains on the rest of the empire rather than centers of innovation, and it became a
rentier state. It might have been able to stave off the Germanic invasions for a while longer with steel weapons- the Han never fell to external invasion, but rather gave way to a succession of smaller, shorter-lived dynasties for a couple centuries until the Tang period.
I think a more interesting point of diversion is one in which the elite of Mecca succeed in offing Muhammed before Islam gets off the ground. Byzantium came very close to re-establishing the old Roman empire under Justinian, and Persia might have had a bit of a rebirth, too. (A major reason why the Arabs were able to take over most of the Middle East with astonishing speed was that Persia and Byzantium insisted on wasting their national coffers on pointless wars with each other).
It's difficult to imagine China taking over Persia- Persia is just way too far. I vaguely remember reading at some point- RedKing may be able to correct me on this- about a Tang-era campaign to modern Sinkiang and central Asia that completely depleted the empire's once-massive funds because 90% of supplies were eaten by the supply lines. Byzantium has a much better shot at Persia since its armies don't have to walk over hundreds of miles of wasteland to get there, but it would probably still be a net minus to the empire. (Maybe...I really don't know. The Achaemenids were pretty successful, and bigger than the Romans, although there had to have been a good deal of internal decay by the 4th century or Alexander would have had a much harder time).