How does evolution take the paths of convergent evolution versus divergent evolution?
Divergent and convergent evolution are quite different, but both demonstrate the flexibility of genetic variation.
In divergent evolution, a group of individuals is somehow isolated from the rest of its species, and over time the isolated group accumulates mutations and variations that make it different from the original group that it belonged to. This can occur because of the founder effect or because of genetic drift, both of which tend to occur more frequently in small groups, or it can happen as a response to a different environment from the original groups'. In divergent evolution, we can see homologous structures, which are features that are similar in two different species because they originate from the same ancestral structure. An example would be the finger-like bones that can be seen inside a whale's flipper, which testify to the fact that whale ancestors had feet rather than fins.
In convergent evolution, we see features that seem to be similar because they fulfill a similar function, but which, upon closer examination, can be shown to have originated from different sources. An example of this would be wings - butterflies, birds, and bats have all have wings, but their wings are all derived from different ancestral structures.
A way to explain it simply would be to say that divergent evolution uses the same materials to build two different things, while convergent evolution uses different materials to build the same thing.