I've done actual experiments in aquariums and such, lol. Keep in mind there's fluoro, mono, AND copolymers. Most tippet materials are copolymers, even if we call them mono. Some leader materials, like Maxima, are truly mono's. But for the sake of using standard terms we'll call the typical copolymer tippets to be "mono".
Visibility - Fluoro is slightly less visible under water. The difference depends somewhat on conditions. The advantage is lost on the surface, if anything mono is less visible there, I think because it floats higher and a smaller diameter is in the actual water. I've looked from underneath in aquariums. So the fluoro for nymphs, mono for dries has some merit.
Stiffness - There are differences between brands of both, and some overlap. But generally speaking mono's are more supple, fluoros stiffer. If fish are drag shy as opposed to line shy, as is the case most of the time, mono will generate more strikes, fluoro will cast easier. You can determine suppleness by cutting a few inches of each to be compared and clamping them in a fly tying vise. Which hangs lower? That's a more supple material. I like my leaders stiff and my tippets supple, personally.
Abrasion resistance - Fluoro better. Again, fluoro for nymphs, as you get less degradation from rocks and such over time. Plus, toothy critters, fluoro = better. I like fluoro for BIG trout, for instance, think steelhead, as well as toothier non-trout species. It's not quite steel wire but it's better than mono for this. If hooked up to a muskie I'd much rather rock fluoro than mono.
Density - fluoro denser than mono. Will get nymphs to the bottom slightly better. Mono will float a dry slightly better. We're not talking worlds of difference here.
Stretch - fluoro stretches less. That's good if tight line nymphing, you're more "in touch" with the fly. For big fish like steelhead though, having some stretchy mono somewhere in the leader, maybe not the tippet, acts as a shock absorber on head shakes and quick runs. I've actually seen people tie in rubber bands for this stretch affect.
In the end, I use mono for 90% of my trout fishing for the suppleness. If I know I'll be nymphing all day I might switch up. Steelhead trips get fluoro. Beach trips get fluoro.