Hot-rolled steel is rolled to its final shape and dimensions at temperatures higher than 1700 degrees Fahrenheit. This temperature is higher than steel's recrystallization temperature, and hence it is easier to roll into the required shape (that is why hot-rolled steel is cheaper than cold-rolled steel, almost half the price). After the steel cools down, it undergoes some shrinkage, so the final dimensions will not be as precise.

Cold-rolled steel is rolled into its final shape and dimensions at much lower temperatures, essentially, after recrystallization has occurred. Since the steel has already undergone shrinkage, it can be rolled to its final shape with more precise dimensions. I am not sure that, but I think cold-rolled steel has higher strength because it undergoes strain hardening.

Hot rolled steel is formed while at a high red or orange heat (1500°f-1800°f). It allows it to be formed faster and into more drastic shapes. The downside is that the hot steel cools and shrinks, so size tolerance has to be somewhat loose.

Cold-rolled steel is finally formed at a black heat (roughly 400°f - 900°f). It requires more pressure on the rollers and is slightly slower than hot forming. Still, since the steel is at a black heat instead of being above transformation temperature, and the shrinkage upon cooling to room temperature is minimal. Thus, cold-rolled steel will generally have tighter tolerances.

In theory, cold rolling also works to harden the surface layer of the bar or sheet. In most steel alloys, this hardening is rather minimal, and of no practical benefit. However, some alloys are more responsive to respect and gain noticeable hardness at the surface due to work hardening. That is beneficial in some applications requiring a callous part but with better wear resistance than standard structural steel.

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