Strictly speaking, E.V.O. Search For Eden isn’t a dinosaur game — the game allows you to play as all kinds of creatures. Still, there are few titles that utilized dinosaurs as well as this under-appreciated SNES gem.
Some games let you control a character or two; E.V.O. puts you in control of life itself. A mystical manifestation of the Earth named Gaia (like in Captain Planet, but with less Whoopi Goldberg) guides you through a plot that covers more than a billion years. During that time, you slowly influence the development of all biological life.
But some dark force is trying to screw with Earth’s evolutionary development (and it’s probably not creationists, because they haven’t evolved yet). You must stop this evil influence, save the Earth, evolve into a complex life form, and eventually join Gaia in Eden.