It was useful for local news.

E.g., when I saw police helicopter that seemed to be doing a low altitude search pattern about 300 m away, and a TV news helicopter hovering higher up in the same area, I wanted to know if this was something I needed to be concerned about.

A quick search on Twitter and I found a recent tweet about gunfire in a store in that area and a shooter on the loose.

it’s not just big news though, twitter was very useful for hyper local news in cities… to the point that if there were helicopters overhead or some loud noise nearby… usually someone posted about it on twitter - this has been irreplaceable for me, I know less about what’s around me now

Trivial, but salient: sale drops for things like electronics or tickets. The only reason I got Pax East passes in 2014 is because I got a Twitter alert. They were gone 10 minutes later. An email alert showed up a half-hour later. I'm sure people have had similar experiences with more significant events.