In the 1980's I was a member of ACI Committee 544 on "Fiber Reinforced Concrete". I always objected to the characterization of fibers as being "reinforcement" in the classical sense and felt that it led to confusion. It does.
The arguments back then in the formative years of the ACI recommended practice were three-fold. There were the ductile steel fiber producers (U.S. Steel/Mitchell Fibercon), the stiff steel fiber producers (Dramix, etc.), and the glass/poly fiber producers. Glass dropped out rather quickly for general commercial use. More specialized.
Each touted its own benefits. For the poly fibers, it was the number of fibers (many, many thousands). For the ductile steel, it was strain compatibility and finishing. For the stiff steel it was strength and bond.
As hokie66 noted, fibers enhance some of the properties of concrete, but do not provide mass tensile reinforcement as would rebar or wire mesh. For instance, most of my experience (though I've used each type)has been with ductile steel fibers in concrete and asphalt, and I have seen the compressive strength and modulus of rupture of the steel fiber modified concrete increase by as much as 50 percent. About a 10 to 20 percent increase was common for both properties at the recommended "dosage" rates. Increases in strength for poly fiber concrete are gained, but not as much. These strength increases may reduce the required steel in the same manner that increasing the concrete strength by any means would do, but they don't replace the tension steel.
Crack propagation properties are changed quite a bit by fiber enhancement. Steel fibers tend to reduce cracking. Poly fibers tend to increase the space between cracks but make the cracks wider.
You can replace wire mesh in a slab on grade with fiber if you choose...since as a practical matter, you don't really need either. (OK>>>here come the arguments!). DO NOT EVER replace tension steel in a flexural member with fiber. That's patently dangerous.