I use quads as much as I can in a mesh, and switch to triangles only when the quads fail my intended use. This is true whether I'm working in 2D or 3D animation and the reason is that quads are generally easier to work with and allow cleaner polygon flow. (The latter is more significant when modeling 3D meshes but it can apply to 2D meshes as well.) Using quads or tris is not an either or situation either, you can mix both in a shape as appropriate.
As SimplSam noted, it depends on what you want to do with the mesh.
The best way to learn when to use which polygon type in a mesh is to work with it. After some experimentation, knowing when to use what becomes second nature.
General tips about mesh warping:
A Perspective Warp, or mesh made of a single quad, can be used for
true perspective warping. This means the image can be distorted using the rules for perspective drawing. Once you attach another point, it loses this quality, as seen in the image below...
In the first two images, this most noticeable in the vertical spacing of the planks. The one using Perspective Warp is distorting the image properly along an unseen perspective grid, and the regular Quad is spacing the planks in a regularly spaced grid.
As a bonus, I checked to see what happens when I tripled the shape (the third image)...and it totally broke. So, while triangles can display polygon faces more reliably, they might not distort the image as predictably as a quad.
A mesh with multiple quads can be used to deform an image in complex ways. Quads are the easiest and cleanest mesh to work with. However, when a quad is distorted too much, it can become non-planar and disappear or distort an image unpredictably.
A triangle cannot become non-planar so a mesh made of multiple triangles will display reliably. However, when distorted too far, triangles can distort the image unpredictably.
You can mix quads and triangles in the same mesh to get the best qualities of both polygon types.
Images are not distorted along curved lines, only along the straight edge between two points. If you need to distort an image along a curve, you need to add more polygons to reduce the faceting along the edges.
When creating a complex mesh, I like to duplicate my pre-converted vector layer and hide it, so I have a backup layer to regress to if I decide to change the mesh structure or polygon type.
Hope this helps explain why we need each of these different types.
