The basic mistake here is thinking that mutation is a crucial step in the creation of species, or that species are created in a single step at all.
Speciation, like most things in evolution, is usually a very gradual process. It can be exceedingly difficult to tell when one form has become “different” from another to be considered a new species. This is demonstrated by constant disputes and differences of opinion over which populations represent subspecies of the same species, and which are different species.
The general consensus is that speciation typically takes place through a process called allopatric speciation, that is, via populations that live in different places. There has been some controversy whether sympatric speciation - that is, speciation within a single population - can occur at all. The general view is that it does occur on occasion, but is rare, at least in animals.
Typically what happens is that two populations become isolated from one another, and no longer exchange individuals through dispersal. This can happen, for example, if climate changes, and populations of a forest-dwelling species become isolated by the development of a intervening large area of grassland; or if a mountain range arises and blocks migration. It can also happen through dispersal, when wandering individuals colonize an offshore island or an isolated mountain peak.
Once populations become isolated, they begin to differentiate. This will happen even in the absence of mutation. If a population has been established by a small number of founding individuals, it will often have a different genetic makeup just by chance. This is known as the “founder effect.” Also, while the population is small, chance events can have an effect on its genetic makeup, even in the absence of selection. This is known as “genetic drift.”
Beyond this, isolated populations will very often be living in different climates, with different food sources, and different competitors. They will undergo natural selection in different directions due to these factors, which will also change gene frequencies.
All of these factors can cause populations to differentiate even without mutation. Of course, over a period of time, different random mutations will accumulate in the two populations and further serve to differentiate them. Over time, the two populations will diverge enough to become recognizable as separate species.
If climate or other factors change, two such isolated populations may come in contact once again. If they haven’t differentiated very much, they may interbreed, and will be considered subspecies of the same species. If they have changed quite a bit, they may be unable to interbreed, or do so rarely, and will be considered different species.
There’s more to it than that, but that’s the outline.