I know what you mean about the annoyance of stubborn facial hair!
Depilatories are difficult to use effectively, especially on the face, and especially on coarse hairs. Usually you'd have to leave the chemical on so long that you burn your skin as well as dissolve the hairs.
Some depilatories can be useful on fine, wispy facial hairs, but many should be limited to less sensitive skin like legs. Every time you use a depilatory, you should to a test patch as directed, even if you've used the same brand many times.
If you don't have a ton of hairs, plucking is the most popular option, since waxing, threading and sugaring can take off all the hairs, even the downy ones. These hairs actually make the skin look a little softer, and without them your face can look a little waxy.
Some people use the rotary epilators made for the face, but this is painful to some, and again, it can take off hairs you may want to leave. That's why plucking, although time consuming, is the most precise way to get only the hairs you want.
If you want a permanent solution, there's electrolysis (pro or home) and laser (if you have light skin and dark hair). These have benefits and drawbacks, so I'd read up what I've got to say on hairfacts to help you decide what's best for you.
Hairfacts: electrolysis Hairfacts: home electrolysis Hairfacts: laser Hairfacts: tweezing/plucking