Copolymer – Synthetic, mostly including nylon, polymers and resin. Usually quite supple with reasonable stretch and quite strong for the diameter. Best suited for situations where thinner softer material allows delicate presentation including when slack line and dead drift are required. Low density means it is less likely to sink and drag the fly under. Good for dries and small nymphs when targeting spooky fish in clear water.
Fluorocarbon – Polyvinylidebfluoride(PVDF). Good abrasion resistance and heavier than water so will sink. Ideally suited to nymphs, streamers and other sub surface flies, but also with dry flies if you want your tippet to sink out of sight. Some pre-stretched fluorocarbons are finer for the breaking strain, but can result in weaker knots.  Fluorocarbon doesn’t deteriorate.