The theory of evolution is a theory that explains that something changes with time.
It also explains how something that used to be one and the same species split into two separate populations, each of which then gradually evolved increasingly different from each other with time. It thus entails both that there should be similarities but also differences.
This idea predicts transitional forms between different groups of organisms that share common ancestry. Transitional forms that there is no reason to think should exist if there was never any gradual transformation of species from their common ancestors.
Also, the particular types of differences can be predictions of common descent. For example, transition bias. Since biochemically we observe most mutations to be transitions (A<->G, or C<->T), if common descent is true then species that share common ancestry should have most of their DNA differences correspond to mutations between A and G, or C and T. We find that.
As expected from the biochemical causes of mutations, the majority of differences both between individual humans, and between the human species and other primates, consist of transitions. Fantastic corroboration of the theory of common descent of primates.
A predicted pattern in differences and similarities can in that way be evidence for a historical occurrence.