Roman Empire? Who cares. The people of the Roman Empire? Much different.
The bicycle would have been the simplest modern technology to imitate in antiquity, and it alone would have made life safer and more efficient. It would have also made slavery harder to enforce. It's just surprising that nobody invented it until modern times.
The bicycle is deceptively simple. First off a Roman bicycle is going to be heavier, simply because it'll end up being made mainly of wood with expensive metal used only where absolutely necessary. Imagine something like an early velocipede only with pedals and a chain drive. The wheels in particular will be heavier for there will be no wire-spoked wheels. Also because of lack of materials, it'll have to have solid tires rather than pneumatic ones, so it'll be far less comfortable and the lack of rubber means they won't grip as well making them less safe. Last but not least, you can forget about there being ball bearings for the wheels. Maybe roller bearings or a plain bearing.
Given the expense involved in making even the inferior Roman version outlined above, dropping a bicycle off in the Roman Empire is not going to set off transportation revolution.
