cutting out the design and sewing it on which always looks terrible..... Most times, I've seen the outcome turn out shaky, unevenly painted and just unprofessional looking altogether.
This is because it is done by inexperienced folks who didn't practice enough before trying the technique on their costume. The annoying old adage "practice makes perfect" is very true, the more you practice the better your results.
Personally, I like applique a lot, I find it's easier to get the hang of through small practice pieces faster than painting. With painting, there are a lot more variables, and using cheap paint or crummy brushes can hinder the final appearance of an otherwise steady handed technique. In applique, if you are sewing around your shape and the fusible web peels up off your base, you can stop, go iron it down again and return to sewing. In painting with a stencil, if the stencil moves it can screw up everything in a way that you can't fix easily.
But it really depends on the costume for which will be the best technique.