Mathematician/computer programmer's answer here:
There is a continuum of different animals — in fact it's pretty fair to say that every animal occupies a different place on this continuum. They're just not uniformly distributed over the continuum; they're clustered around forms that are most likely to survive and reproduce, and the lowest-energy paths between them.
This is because evolution is basically a stochastic optimization algorithm, one that finds the "best" set of parameters for maximizing some function by randomly perturbing an initial set of input values. In fact, some of the best optimization algorithms today are based on ideas drawn directly from evolution and called "evolutionary algorithms".
In mathematics, given enough time, these algorithms will all converge on one optimum solution and nothing else. Why isn't it like that in nature? Because in mathematics, the "fitness function" that we're optimizing for stays the same for a given problem for as long as the algorithm runs. It represents the specific problem we're trying to solve. In nature, there's no outside force imposing a fitness function; an organism's survival depends on millions of factors in its environment which change over time, many of which depend on the survival and properties of the other organisms in its environment (competition, symbiosis, predator/prey relationships, etc.) This is a chaotic system so complex that it could easily go for billions of years without reaching a steady state, and even if it did, external changes (like the odd asteroid impact, to use an extreme example) would still come around to shake things up.
Another reason for the clustering is because a lot of the "intermediate states" in the genetic space have a fitness of zero — these are the states between species that can't interbreed, or that have sterile offspring. The categorization of "species" is based on this, and although it's not exact, it's still generally true that the more different two creatures are, the less likely they can have viable offspring. This makes evolution more likely to explore the spaces near already-successful organisms, and less likely to produce radical new things by hybridization.