Safe Upgrade looks great, and has some features that we use on a regular basis for clients whose WP sites we maintain. However when we're testing our updates/tweaks, we do so on a staging site then run a comparison using Wraith: https://github.com/BBC-News/wraith
It allows us to set 2 sites (live & staging), a list of page URLS, and a list of screen sizes to compare (and more but this is all I use!)
The 2 sites wouldn't be necessary as it looks like Safe Upgrade already does a 'before & after', but it'd be great to be able to get a comparison for more than just the homepage (shortcodes, plugins etc. only being used on specific pages) and to be able to test on mobile device screen sizes too!