Species evolve independently possibly due to geographical isolations or behavioural isolations.
Geographical isolation includes the isolation of 2 groups of the same species. Since these 2 groups live in different locations, (e.g. a volcanic eruption resulting in a barrier between one side of an ocean and another side of the ocean), they will have different selection pressures in their different environments as well (e.g. one side of the ocean may have more sunlight and thus more underwater plantations than the other side of the ocean). Natural selection will eliminate those with disadvantageous characteristics (e.g. Fishes that only eat plants and nothing else on the side of the ocean with little plantations) with unfavourable alleles, and select for those with advantageous characteristics (e.g. Fishes are able to eat plants and other organic substances as well on the side of the ocean with little plantations) with favourable alleles.
Since the 2 groups have different selection pressures, natural selection will occur in different ways, selecting for and against different types of fishes with different types of alleles. Also, because of the barrier, they are not able to mate with each other, and there are no mixing of genes from one side of the ocean and the other side. They are genetically isolated. As genetic drift occurs over time, their gene pools become different from each other. Thus, they evolve independently.
Hope this helps! :)
Cell division or reproduction process is called mitosis
