What are the pros and cons of each type of drywall joint tape? Are there situations where one should always be used over the other?
Nylon mesh tape has some pros and good applications. It is best used on joints that may have more tendency to crack, seams with wider gaps, boards that don't fit real flush to each other. It is much stronger than paper tape, but more difficult to cover using straight tools.(as opposed to beveled or bow trowels) It saves time as you do not need a base mud coat, the tape adheres to the wall by itself. The first coat should be a bit thicker than you would use with paper tape, thus taking longer to dry.
Paper tape is preferred by pros because it is lighter, thinner, cuts easier and much cheaper. When using paper tape, the base coat and first overcoat should be a setting type mud, not general purpose. Not absolutely necessary, but setting mud is harder, holds tape better and cures quickly. Usually a 45 or 60 minute set up type works great. The paper tape must be completely saturated and embedded into the base coat or air bubbles will appear. This first coat should be very thin and firmly troweled over with a 6 or 8 inch stiff knife to push out all air, make good contact with base coat and remove excess base coat mud. Subsequent coats of mud can then be the GP premixed stuff.