The short answer is that they are very similar, but not exactly the same.
In fact, if you were to look them up in the dictionary (links provided below), you would find that they have very overlapping definitions and might seem to be complete synonyms. However, there is a rather subtle difference in how they are used in every day conversation, with each of them generally favoring one of the two common definitions over the other one.
Usage tends to refer to the proper way to use something, or perhaps more generally, it is said when discussing how to use something, properly or improperly.
Merriam-Webster defines it as "1. firmly established and generally accepted practice or procedure; 2. the action, amount, or mode of using."
Use tends to be more general; it usually refers simply to the act of using something, outside of a discussion of proper or improper usage.
Merriam-Webster defines it as "1. the act or practice of employing something; 2. habitual or customary usage." (There are others as well.)
For example, in the sample sentence you gave, "The usage of chemical weapons" sounds like the beginning of a sentence on how to use them, such as "The usage of chemical weapons is a very dangerous topic.", whereas "The use of chemical weapons" sounds like a phrase from a political discussion, such as "The use of chemical weapons in this manner is a crime against humanity."
If you were to say "The usage of chemical weapons in this manner is a crime against humanity," it would sound like you were critiquing their form, rather than saying that the act of using them is itself wrong.
I hope that makes sense. These are not hard, fast rules, and, like everything in language, there is some overlap in meaning.
In summary, usage refers to how something is used, and use refers to the act of using it.