The Application Source is the most recent way the candidate was referred to the job they applied for. If there is no referral id from the site they used (such as Indeed), then whatever they choose from the "How did you first hear about this opportunity?" section will display. (If you have questions about editing that dropdown list, you can find instructions on how to do so here.
The Application Method is how the application was processed into the system. Did it come through Apply with Zip, Apply with Indeed, etc.
Example: A candidate finds a job on Indeed and clicks through to your job board – the source is now Indeed. If the candidate then uses the apply with Indeed button, the method is apply-with-Indeed. If they use your application on your job board (with no integration) then the method is job-board.