The terms have to do with the "thickness" of the voxelization. I'll illustrate with the help of a diagram about 2D line rasterization (from this unrelated question).
On the right is the typical line rasterization: the algorithm finds the one pixel nearest the line within each row (or column, depending on slope). This produces what we usually think of as a "1-pixel-thick" line. On the left is a conservative rasterization, which finds every pixel whose rectangle is touched by the line, and it produces a thicker line.
6-separating voxelization is like the thin line on the right, and 26-separating is like the thick line on the left, but in 3D. If you imagine the line is actually a triangle viewed on-edge, this is analogous to what the voxelization would look like.
Different types of voxelization may be better depending on what you're going to do with the voxelized data later. If you're using the voxels as a spatial hierarchy to find triangles that intersect a given region, you probably want the thick voxelization, as it's conservative. The thick voxelization might also be preferable for ray-marching, as the thin voxelization could be missed by a diagonal ray. On the other hand, the thin voxelization is a more faithful representation of the original surface, which is probably better for visibility tests, collision detection, fluid simulation, and suchlike.
The "n-separating" terminology is a bit unfortunate, but here's what it's getting at. Imagine you're doing a 3D flood-fill in the voxel grid, but in the flood-fill you only look at the 6 direct neighbors of each voxel (±1 step along each axis). Then the "6-separating" (thin) voxelization will stop the flood-fill: it suffices to separate the two sides of the surface, if only 6 neighbors are considered. On the other hand, suppose your flood-fill was allowed to go to diagonal neighbors as well—26 neighbors in all (3×3×3 neighborhood of voxels). Then the 6-separating voxelization wouldn't stop the flood fill, but the 26-separating (thick) one would.