What is the difference between BiLinear and TriLinear

It bothers me, is one of them faster to render textures with, which is better in cases of Performance and Quality?

The difference between bilinear and trilinear is that trilinear blends between mipmaps. Clearly it takes more time to do something than to do nothing, so trilinear is necessarily slower than bilinear. If you don’t have mipmaps (e.g., for typical GUI 2D usage), then trilinear is useless.