The Romans did not invent the heavy-armoured horseman: on the contrary, the innovators were the Assyrians, whose monuments uniquely illustrate the evolution of cavalry technique in antiquity. [link]
Cavalry with lances/spears could be formidable: swift and deadly. We have a detailed contemporary description by Julian the Apostate (331 - 26 June 363). Julian was a nephew of Constantine who became emperor; he was called "Apostate" because he rejected Christianity and turned back to Greek beliefs. He wrote:
...their limbs were fitted with armour that followed closely the outline of the human form. It covers the arms from wrist to elbow and thence to the shoulder, while a cuirass made of small pieces protects the shoulders, back and breast. The head and face are covered by a metal mask which makes its wearer look like a glittering statue, for not even the thighs and legs and the very ends of the feet lack this armour. It is attached to the cuirass by fine chain-armour like a web, so that no part of the body is visible and uncovered, for this woven covering protects the hands as well, and is so flexible that the wearers can bend even the fingers.
The horse was the weak point, however, since charging into a crowd of enemy soldiers risked a sharp blade to the horse. An excavation in Syria at Dura Europos, a Roman border city above the Euphrates, uncovered two iron/copper horse armors with some of the fabric backing still attached. Besides those rare examples, we have a few carvings showing samples of the equine armor. (The illustration gives an Assyrian example.)
Let's look at what the later Middle Ages did with cavalry next.