What is the difference between linear and plane polarization?
I don't think there's any difference. Linear polarization is the more modern and, in my view, better term.
The idea is that in a linearly polarised wave, if we represent the displacement vector at a point in the path of the wave by an arrow, then the arrow tip oscillates back and forth along a straight line. [For circularly polarised light, the tip of the displacement arrow goes round and round a circular path.]
The term 'Plane polarised' probably comes from a linearly (!) polarised transverse wave in a rope viewed along the rope, so that the displacements all lie in one plane (a plane containing the rope itself). This is fine, but we don't call a circularly polarised wave 'cylindrically polarised' or 'helically polarised' do we?