Using random UUIDs as an invoice number wouldn't cut it, you'd have to have a sequential ID in some form. You could of course have both but that's adding unnecessary complexity.
Another example, if you are designing a website and lets say give each blog post a UUID, a URL of:
website.com/blog/my-post-34
Looks better than
website.com/blog/my-post-d173affb-6bb8-4435-bef1-4e29409dff4c
Unique sequential Id's also give some clues about the relative age of rows (eg ID 100 is older than ID 50). This doesn't come up often but can sometimes be useful.
Regardless, I always find using a unique int ID for pretty much every record makes your job 10x easier and I fail to see why you would ever want to do anything different. If you want to 'hide' the ID in special cases (eg order numbers) better to generate a unique random int ID.