So just to be clear mono, copolymer and fluoro are all monofilament, in that it is all a single extrusion. However we anglers usually like to define them separately so for the purposes of this post that's what I'll do.
Mono & copolymer: Basically the same stuff. A relatively inexpensive tippet material that has good stretching properties, now whilst this makes for good shock absorption and less risk of breaking it can reduce sensitivity and bite detection. It tends to be supple and not to have memory problems and lies straight. Diameter to strength can be excellent with better brands having surprisingly fine diameters for their breaking strain and good abrasion resistance. Mono and copolymer are not UV resistant and will eventually break down in sunlight.
Fluoro: I've got to say there are some claims made of fluoro that I don't believe and that do not stand up to scrutiny.
Take a length of each material of the same diameter and fluoro will be heavier than mono, however the claims that it sinks and so is better for sub surface fishing is a load of eyewash! If it doesn't break the surface tension of the water fluoro floats just like mono. If it breaks the surface tension fluoro and mono sinks, however fluoro does sink more quickly but if fishing a wet fly or nymph it doesn't help to sink the fly quicker.
I've read how fluoro is near invisible underwater due to its refractive index, mmmm, right!! Try this little experiment for yourself - Take a length of each material of the same diameter and place them in a glass of water, both are clearly visible. Now drop the same pieces on to the water, both will stick in the surface tension, finally poke them under the water and both will sink!
Fluoro is generally stiffer and less stretchy, this is good for bite detection but bad for snapping and memory retention. Some claim knot strength is not so good, but as long as appropriate knots are well tied I don't think there is a problem, but because it is stiffer and less flexible it may contribute to knot breaks.
Finally fluoro is UV resistant so does not break down in sunlight, this one fact is why I will not consider using it. If this stuff is left lying around it's there forever, it will not degrade!
Fluoro is a good deal more expensive than mono so you have to decide if it's worth the premium.
So to conclude: there are good and bad points with both materials, it's up to you to decide which is best for you. Personally I use and recommend good quality mono or copolymer.