Perlin Noise: The Near Universal Texture

Simplex noise is arguably inferior in quality to improved perlin noise.the difference usually doesn't matter. It's also faster, but on an order of difference that usually also doesn't matter.

All of the other noise types can give you different things for different purposes. Most of them look pretty similar, but there are differences! for example, value noise is quite similar to perlin noise (which actually interpolates gradients instead of values). Some of the popular perlin noise tutorials actually teach value noise instead accidently! What that means though, is that the low frequencies are unbounded in value noise, because you may randomly have clumps of like values in a row. Perlin noise is guaranteed to have zeros at the lattice points of the noise and so produces band-limited noise that is useful when you're doing a summation.