Cogitate means think deeply about something - maybe the meaning of life. Excogitate means plan - maybe how to achieve your career goals. The difference is that after you cogitate, you will have a deeper understanding of something. After you excogitate, you will have a series of steps you need to take.

Claire has given a great answer: cogitate is to ponder something (think) while excogitate is to plan something (action). I would say both are relatively rare words. Hope this helps :)