Sheath. The sheath's thread can be braided either individually (e.g. Edelweiss' HD Cover) or in pairs (industry standard). Individually braided sheaths will have a smoother feeling and reduced friction going through belay devices while standard braiding is preferred on the more skinny models. More subjective is the argument on durability and abrasion resistance. However, all manufacturers will test the resistance to abrasion and sharp edges of their sheaths.
Core. The industry standard is to use cabled cores. Cabled cores are made of strands of core yarns running parallel inside the sheath. High quality cabled cores paired with high quality sheaths can make the overall feeling of the rope extremely supple and flexible. An interesting alternative is Millet's triaxiale concept. Triaxiale cores are made of strands of core yarns further braided together in bundles of three, thus making the core stronger, less prone to internal abrasion/slippage and allowing the rope to maintain its original diameter for a longer period of time.
Treatments. The sheath is the first line of defence against external agents and this is why most ropes will nowadays feature a basic dry treatment, some manufacturers going all the way to complete dry cover treatment involving the core too (e.g. Tendon's Teflon Complete Shield). Although for rock climbing dry treatment is not as important, for other uses (e.g. ice climbing) it's definitely a more important feature.