Apple - Find Differences between Time Machine backups

The tmutil command-line tool has a compare option that does this. Just give it the file paths to the two snapshot folders on the backup volume, and it does the rest. The paths get rather long, so it's easier to cd to the backup parent, then just specify the snapshots. For example, compare the March 27 2016 10:11:12am and March 26 5:43:21pm snapshots of a computer named "Ipinak's Mac", on a backup volume named "Backup Disk", you'd use:

cd "/Volumes/Backup Disk/Backups.backupdb/Ipinak's Mac"
tmutil compare 2016-03-27-101112 2016-03-26-174321

Timedog is a perl script that will list the files which were updated in a particular backup. It doesn't show files that were deleted, though.

I found Gordon Davisson's answer very helpful.

As I still ran into a problem (on macOS Mojave), I'd like to mention a setting I had to adjust before I could use tmutil compare properly, as it just gave me the following output when I tried to compare two backups:

$ tmutil compare /Volumes/BackupVolume/... /Volumes/BackupVolume/...
-------------------------------------
Added:         0B
Removed:       0B
Changed:       0B

tmutil listbackups gave me the answer to my problem:

$ tmutil listbackups
The operation could not be completed because tmutil could not access
private application data on the backup disk. Use the Privacy tab
in the Security and Privacy preference pane to add Terminal to the
list of applications which can access Application Data.

So if you add your terminal application to System Preferences > Security & Privacy > Privacy > Full Disk Access tmutil compare should work fine for you.