The Pauli principle is not a force in the sense that gravity is not force. Yes, you can distinguish between a "force" and the phenomenon responsible for the force (gravity vs gravitational force). What is the difference between these two statements?
1) That's not a force, it's the playing out of the fundamental symmetries in quantum physics, normally phrased here as the Pauli exclusion principle.
2) There's no force on that falling object in a vacuum, it's just following the geodesic dictated by the symmetries in General Relativity.
PEP is not a force, in the sense that it's not 'dynamical:' it can't actually affect the Hamiltonian/Lagrangian of the world. And it's not a symmetry either, it's a consequence of the behaviour of 'fields' under rotations: see spin-statistics theorem. (Explanation of the field business: modern physics postulates that at every point in space and time there are a certain number of degrees of freedom, and we call them fields and 'quantising' gives us particles - and particles are just spatially localised excitations when you don't look closely at them.)
The re...
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: