Even if two people have the same set of genes, because of the different environments they experience – i.e. in the womb (maybe one got more nutrients than the other), the types of illnesses they have had as they grew up; the lifestyle they lead, etc, there can be differences in how much each twins' genes are turned on or off – this can then affect the way each twin looks and behaves. This effect increases as the identical twins get older, because they are exposed to more and more diverse environmental events (different illnesses, jobs, friends, habits etc).

You can read more about this idea of "identical genes, different twins" here.