This question is something which is asked a lot by a lot of different people, even those who work with tools all the time, and there doesn’t seem to be a very satisfactory answer. Some people say that there is no difference; that the different names developed as a result of Americans calling it something different from the British, while others claim that the name difference is to differentiate between an adjustable spanner (meaning a wrench) and a set spanner (not adjustable). What is clear though, is that they both perform the same function.
Both spanners and wrenches serve the primary function of tightening and loosening nuts and bolts. Nuts and bolts are found everywhere in homes and machinery to secure one component to another, and they are an essential tool for somebody working as an electrician or plumber, among other professions.
If you’re interested in this question you may also be interested in this too, which is an article on the pros and cons of adjustable and non-adjustable spanners.