This may be over-simplifying the process, but to summarize:  The DNN "auto-install" feature (the one you can enable/disable) checks your current DNN version against the version value recorded during the last install/upgrade.  If DNN finds that the current assembly (files) are a newer version than the recorded version value then "auto-install" automatically initiates an "upgrade" attempt.  It may also check the DB version but in practice I've found auto-install will not initiate if my files are up to date but my DB is not (for example, due to a failed previous DB update attempt).  By contrast, auto-install always seems to work when my recorded assembly version (version of files from previous install/update) is older than the current version of DNN files in my site directory.

"Auto-install" is mostly just a convienience feature since you can call the upgrade routine yourself by browsing to the following URL: 

   http://{your site's domain here}/Install/install.aspx?mode=upgrade