There are many reasons to have tighter chem limits than 5CT. The chemical composition requirements of 5CT are pretty loose; they have to be since they are not in the business of telling manufacturers how to make the product. They are just trying to define a standard product.
The effect of carbon is that higher carbon increases strength. At higher levels, it can result in quench cracking (L-80 must be quenched, I believe). Higher carbon can also decrease toughness (how much depends on the process). L-80 also has toughness requirements, I believe.
Manganese increases hardenability. That is, it allows thicker sections to harden on quenching. It also allows sections to be oil quenched and harden that would crack with water quenching. At higher levels,it also can promote banding and lower toughness.
S & P are impurities. Limits of 0.009 S and 0.020 P are common for modern steels that have to meet toughness requirements. API has higher limits because they can't decide at what level you can no longer meet the requirements so the put the limits so high the manufacturer has to decide what level he needs.
Si is a deoxidizer; it lower the oxygen content of the molten steel. It can contribute to hardenability, but usually it is just used for deoxidization. Normal levels are around 0.2%, but even at 0.4, there isn't a lot of problems. ERW steels generally have very low Si, <0.10, as it causes weld defects in the weld seam. For seamless product, it generally doesn't cause any problems.
What is not mentioned are the other allowing elements. Most certainly, there is Chromium and Molybdenum added intentionally, and residual levels of Nickel and other elements. In addition, aluminum (aluminium) is likely to be present in around 0.02% level. Vanadium, Titanium, and Columbium (Niobium) can also be present in low levels to promote grain refinement.
Unless you really know what you are wanting, you are probably better off letting the manufacturer decide on the chemical composition needed to meet the requirements, since he has to control his processes. Otherwise, you find yourself in the position of telling the baker not only how to bake the cake, but how it must taste, too.
rp