Most common I can think of is when I want to read a web page on something bigger than my iPhone. Currently I have to email myself a link, and I lose my scroll position.
Just noting that if you use Safari you can use iCloud tabs to load pages from other devices. I believe Chrome is similar in this regard. Handoff just reduces the friction in even this process, so you don't even need to specify which page to pick up reading on.
iCloud tab syncing has a pretty bad lag. Maybe a minute to update on my Mac. Fine most of the time, but annoying if you want to do a Handoff-style scenario. It also doesn't show your current iPhone "tab" highlighted in any way, which is odd.
So, I'm glad Apple saw it as something to improve.
