For the most part, regular fiberglass mesh tape is "amateur hour". It's used by DIYers and "pros" who are not full time drywallers who don't know what they're doing. There are actually very few circumstances I would use that tape (now that there is a third option, I never use it.)
Mesh tape is weaker than paper tape. Therefore you need to use setting compound with it to achieve full strength rather than standard drying joint compound. If you have to mix your own compound, then where did the convenience of mesh tape go? Out the window.
Paper tape can be more difficult to use for DIYers since it can trap bubbles and form weak spots or high spots that are a problem when sanding and painting. It is strong if done correctly.
Another product called FibaFuse solves these problems. It is as strong or stronger than paper tape, but with almost as much convenience as pre-sticky mesh tape. I recommend it for all taping except for corners (it can be used in corners, you just have to be more careful with it. I still use paper for corners myself.) It's a finer weave than mesh tape, but compound will still flow through it. So once you stick it on with a few dabs of compound, you can tape over it just like mesh tape, if you prefer, without having to lay a base layer like with paper tape. I usually lay a base coat, but you can't really get bubbles like with paper tape.
The second great advantage of this stuff, over both paper tape and mesh tape, is that you can sand it. With paper or mesh, if you sand down to the tape, you're kind of screwed. This stuff is no problem.
Not to be confused with FibaTape - it's FibaFuse.