Can I get a notification emailed to me every time a contact with a certain tag is updated?
I think meeting all these requirements would involve custom code. Some possible approaches to meet some of the requirements...

(1) In Advanced Search, search for contacts with the relevant tag and under Change Log, modified date yesterday. Save as a smart group. Then set up a Constituent Detail report filtered by this smart group. Have it emailed daily.
This will not give you the contacts' previous data.

(2) Enable logging, at Administer -> System Settings -> Misc. Use the Contact Logging Report.
This does not allow filtering by tag. However this feature could be added, with code customisation.

(3) A custom solution using the civicrm_pre hook to provide real-time notification of changes. This could include before-and-after values.

(4) A custom solution running under cron, using Civi logging to check for changes and retrieve previous values.