Mesh works with triangles. A triangle is a 3 sided polygon. Poly works with polygons, with any number of sides. Internally, each polygon is still made from triangles, but the trianglesness is hidden from the user. Instead of seeng a square as 2 triangles, they see it as a square.
Read the help file? OK... don't jump on me. Patches are polygonal surfaces that you control by moving around their edge-splines. They have their benefits and drawbacks. Most people don't use them on game assets, although Scott likes them (or did... Scott?).
One thing we've found useful (well, the only thing really) with Edit Mesh is that the inner edges of a polygon stay fixed, which is good when we need to export morph targets to the game. Edit Poly tends to re-arrange those inner edges to the shortest lengths, as polygons are squished and stretched during modeling. This is…