Theoretically, the only difference between hot rolled and cold rolled steels is that hot rolled steel is rolled to its final dimensions while hot enough to scale (over about 1700 degrees F) while cold rolled steel is rolled to its final dimensions well below scaling temperatures.
So----- If you are making ½” square HOT rolled steel, you have to estimate what the final size will be after the product cools, whereas you can finish the Cold rolled steel to much closer tolerances right in the sizing rollers and that is what you get.

There are some other things to consider, too:
-The finished tolerances on hot rolled steels are looser than on cold rolled. Not only the plus or minus tolerance from nominal size, but the "square-ness" of the product. And, I can tell you from personal experience that there’s a lot of trapezoidal (HOT rolled) A36 out there. So, if you need a specific size and you are going to go to a “surplus” place, bring your ruler, square and micrometer to make sure you get what you need.

- I have been told that, in order to get the cold rolled steel to come out with a nice finish, they might use "cleaner" ingots from which to roll the product. This means that you’d get fewer slag or carbon inclusions with cold rolled steels.

-Note that I haven't talked about the chemistry of the steel at all. You can get cold rolled or hot rolled 1045 and you can perform either process on C1018. But since we often talk about using "mild" steels, the two steels that we end up having around most often are C1018----which is quite often sold in cold rolled form and A36 which is always hot rolled.
-One other difference that may be of interest to the blacksmith is that if you buy "1018”cold rolled steel", you can be pretty sure that it has close to a 0.18% carbon content and few other impurities. But the spec for A36 can let the carbon content go as high as 0.29% and it can contain many more impurities. More carbon makes it harder...

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