Drywall installers have flat and curved trowels at their disposal. Some controversy exists within the drywalling community as to which trowel is better as some installers don't even use curved trowels because of the difficulties they present. Others find the curved trowels useful for finishing off some types of seams between drywall sheets.
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Curved Trowel: Butt Joints
When two drywall pieces come together, some pieces have slightly recessed edges that are easier to line up and level with a flat trowel. However, a curved trowel does not apply as much direct horizontal pressure as a flat one, and so finessing out a level seam between two sheets with a curved tool is easier.
Curved Trowel: Intermediate Coats
If you're putting on three coats of drywall for a new construction finish or remodel project, you may want to use a curved trowel for the second coat. The first and third coats must sit flat and level, but the second one can allow some air into it to help the first coat cure properly. A curved trowel allows you to expand that second coat somewhat, adding breathability. After it has dried, you can add a third coat and flat-trowel it to level it.
Flat Trowel: Leveling
Flat trowels are easier to level sheets with (with the exception of finishing up butt joints). Their surface is already level, so you don't have to come behind and finesse it if different ridges have formed. You also don't have to perform as much testing of the walls after you've finished spreading them with a flat trowel; if one section is level and you've used the same technique, the rest should be level to match the surface.
Flat Trowel: Easier on the Coat
If you're moving a curved trowel over an existing coat of drywall, the curved edges can pokes holes in your existing coat, threatening the integrity of the whole sheet. Flat trowels can also gouge coats, but keeping a flat edge away from a flat surface is easier than it is with a curved edge.