A closed job is one that is complete, i.e. you have hired someone and/or the position is no longer open. You can access all closed jobs by selecting the Closed option at the bottom right of the table on your Dashboard.
Archiving a job lets you keep it around in case you ever want to go back to it in the future and is intended for those jobs that you think you won't need to revisit but are keeping around for record-keeping purposes or "just-in-case". Again, all jobs with the status Archived can be viewed by clicking the Archived option.
The main difference between the closed and archived jobs is that only archived jobs can be deleted. This is to prevent the accidental removal of jobs from your account. Generally we recommend archiving jobs and deleting them only as part of a regular record management process.