If you are talking about traditional black/white double mats, the 8 ply has a higher end look. Museum quality mats are usually solid core, but the 8 ply adds depth. double 4 ply museum that is solid core white or black also looks great, but side by side the 8 ply to me looks nicer.
If you are opting for non traditional black or white, and instead using typical color matching like many frame shops, 4 ply comes in significantly more color options.
As mentioned 8 ply is tough to cut, some mat cutters can’t handle it too well.