Please check your facts before writing nonsense. Mongols used siege engines regularly during their battles, even going so far to spare engineers from opponents and assimilating them into their armies. They also placed huge importance on battlefield organization and tactics (they protected their generals well and used flags for battlefield communication), and they promoted leaders and generals based on merit, not birth. Their armies also required little logistical support, which gave them incredible speed when needed (armies could advance hundred kilometers or even more in a single day), and they placed heavy emphasis on scouting.
As for Roman heavy armor, mongolian composite bows are second only to English longbows in range and draw weight, they could pierce anything except plate armour, which wasn't exactly common (also, lorica segmentata was never comon in Roman legions, regardless what you usually see in movies and books, lorica hamata or squamata were common armors of Roman legionnaires).
As much as i love Roman Empire, in this case it is absolutely no contest, Mongols would win by a landslide.