What is the difference between the TSA and FAA in terms of what they do with airline passengers?

Question

what are their distinct functions or jurisdiction when it comes to dealing with airline passengers and the security of the flights?

Answer 1

The Transportation Security Administration was specifically created because of 9/11 to protect the nation's transportation but the only aspect of it right now is the screening of passengers and cargo at airports.
http://www.tsa.gov/public/display?theme=7

The Federal Aviation Administration has been around almost since commercial flights. It's primary job now is the safety of the planes, how they are flown and taking care of air traffic.
http://www.faa.gov/about/mission/activities/

Basically, it looks like TSA deals with the passengers; FAA deals with the planes, pilots and airlines. They both work for the security of the flights.

Answer 2

Two separate missions.

TSA is responsible for security in all transportation systems, not just aviation.

The FAA is responsible for aviation safety -- pilot qualifications, flight rules & procedures, etc.