Speciation is the formation of new species from pre-existing ones.
Usually by isolation of a population/gene pool.
Eventually gene pools are so different that interbreeding cannot occur.
Allopatric/sympatric speciation occurs.
Divergent evolution is where different populations adapt to different ecological roles / adaptive radiation / avoid competition.
Result of natural selection / competitive advantage of one population over another.
Example, e.g. Darwin's finches / mammalian pentadactyl limb / any other verifiable example of divergent evolution.