Keep in mind I'm not a native speaker of English. Contrex shows one use of "catch up" that I wasn't aware of, so there may be more ways to use "catch up" that I don't know.
"Catch up on" - as far as I'm aware - generally precedes things or tasks. For example: The president has to catch up on sleep. Sleep is the task the president has to catch up on. "Catch up with," however, is generally used to describe encounters between people. For example: The president has to catch up with his daughter, as in he needs to be filled in on the details of her life that have happened since they last spoke in earnest.
"Catch up with" can also be used to express a need to reach the same standard as someone or something else. For example: The university needs to catch up with times. This example suggests that the university is old-fashioned. Another example: The US diving program needs to catch up with China's if the Americans are to contend for the gold medal at the Olympics.
Hope this helps!