what is the difference between employ and employment

There's no difference between the two. Employ is just an archaic form of employment. That said, the verb form of employ is the same as the archaic form, so to avoid confusion, it's better to use employ as a verb and employment as a noun.

Employ is a verb.
Employment is a noun.

For example

I would like to employ (verb) John.
John was desperately looking for new employment (noun) after having been laid off.

You can use employ as a noun, such as in this example: "John found himself in his brother's employ." However, that type of use is archaic and quite uncommon; I would recommend against using it as a noun so as to avoid confusing people.