Pure mono - not many of these, but they are very stiff, and stretch a lot. I like this in my butt sections. Maxima Chameleon is one example.
Flouro - They market "invisibility", but IMO it's B.S. That said, it has some advantages. It does sink better, which is good for using nymphs. It also has far superior abrasion resistance, which means you don't muck it all up rubbing against rocks and sand, or fish's teeth! It doesn't degrade, you can keep it for years, wet and dry cycles, expose it to UV, etc., and it won't go bad, whereas mono will. On stiffness, it's stiffer than most copolymers and more supple than a straight mono. It's often slick, so if I use shot when nymphing it tends to slide on me.
Copolymer - most of what we call "mono" is actually a copolymer. It is the most supple of materials, giving you your best drag free drifts for dry flies. It has some stretch, which is good for fighting fish, you can get away with slightly smaller tippets because there is some "shock absorption". It tends to float, which is good for dry flies. It does do poorly in abrasion resistance, so not good for especially toothy critters, or rubbing against rocks all day long.
Of course, in any of those categories, there's a "range" between different brands. Tippets made for fly fishing will outperform most lines meant for spinning gear. Same stuff, same processes. Yes. But tighter tolerances. You pay more too. In the end, it's not that expensive. I spend less than $20/year on tippet and leader materials. Fluoro will cost considerably more per spool. But I use a low enough amount of it, and it lasts long enough, that a spool will last me years.
With the exception of a leader designed solely for nymphing, I go with a stiff mono butt, then about halfway through the taper, I switch to as supple a copolymer as I can find. That's what I like. It turns over well, and you can adjust the drag/accuracy trade-off with changing tippet size and length.
My dedicated nymphing rig is a short mono butt followed by a long distance of flat fluoro. No real taper.