I suggest you try to use a combination of 2 approaches, as further detailed below.
Approach 1 - Prior approval required
If you'd use the the Profile 2 module, then the Profile 2 Moderation module, which depends on the Profile 2 module, enables moderation on user profiles. Here is a (self explaining?) screenprint from its project page.
Be aware however that, as mentioned on the project page, it requires a Profile revisions patch.
Approach 2 - After the facts
Use the Rules module to be informed about all sorts of updates that you're interested in, similar to what's detailed in my answer to "How to send an eMail notification when a specific field of a node or user has been changed?".
With this approach, and as per all the great things you can do with this module, your options are virtually unlimited. This approach also allows you to implement this type of moderation using a granular approach: start small, maybe only using a few custom rules, and keep refining them and continue adding new ones.
Not to forget all the available integrations with tons of other modules, of which the Flag module is probably your very first priority to also add that module for what you're trying to implement. Just a basic usecase about this:
- Use Rules to flag any user for which some profile update happened.
- Use Views to build a list of such flagged users.
If you'd only implement these 2 bullets of this approach (a matter of minutes, say an hour at max), you'd already have a Version 1.0 of your User Profile Moderation Dashboard (call it your MVP if you prefer ...).
If you're not familiar with Rules, checkout the video tutorials Learn the Rules framework. And/or the similar set of 8 video tutorials about the Flag module.