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Minnaert (more precisely, the Minnaert-A mode or implementation) was designed to replicate the shading of the moon (but not the only one), so it's often called a moon shader. I believe Gaffer uses a similar model when you use its Diffuse Roughness setting. LightWave's own Diffuse Sharpness (originally called Sharp Terminator) is also an attempt to try to simulate this. Minnaert's good for anything that's got a particulate or porous surface, like the moon, which is covered in dust. These surfaces can cause a lot of light to back-scatter. Good examples are metal oxides, brushed metals, dust, stone (particularly sedimentary ones like sandstone, shale, slate, and to a lesser extent, marble and granite, which would be better done with Lambert), unfinished wood, matte paint (dry of course), and anything else like that. If it has a fine rough surface or feels powdery, then Minnaert's your shader.

Minnaert B, the second option in the Minnaert shader is better for fibrous surfaces, especially where the fibers tend to be mainly perpendicular to the surface, like velvet, velour, or even carpets. It's pretty close to Oren-Nayar, which is frequently called a velvet shader. In fact, these two shaders are frequently interchanged with moon and velvet shader, where Minnaert is also called a velvet shader and Oren-Nayar is the moon shader. There's a lot of variations of these, and most other shading models, and the line between them can get a bit blurry.