Explain: Update siteurl and home as well?

I’ve been testing a site migration process and ran into a few weird glitches.

Even though I’ve uploaded my database and wp-content folder to the new host at a new url there are are several items that point to the old url.

For instance the My Sites menu pointed to the former location rather than the new location until I checked “Update siteurl and home as well”

Images that are uploaded and in the system are still pointing to the former url within the page content.

Example:
New location: http://pitzer.wpengine.com/app/using-pitzer-parent-app/
Former Location: http://pitweb.pitzer.edu/app/using-pitzer-parent-app/

Any suggestions as to why media files are still pointing to the old url?

Eventually I’ll need to change the url to http://www.pitzer.edu so I’ll likely run into the same problem again when our entire site migration is complete.