Some examples:
1. Some Saturday Night around midnight try searching Twitter for SNL. (You don't really have to search, it's usually a trending topic). You will instantly find out who the guest is, if there is anything particularly funny, if the monologue bombed, etc. Go ahead and say "SNL is never funny," but it is just an example of real time feedback. You can do the same thing during awards shows or sporting events.
2. Since silicon valley has a relatively high density of Twitter users, I will sometimes find myself doing a twitter search for the road I'm about to take if I'm concerned about traffic. Sure, I can see traffic on Google maps on my iPhone but it only covers freeways and is not as fast or granular as people complaining that they aren't moving for whatever reason at a paricular intersection.
3. Speaking of iPhones, Twitter search was AWESOME for figuring out where lines were long or short. On the days following the 3G launch, doing a search for 'iPhone line' would produce results like "About 200 people in iPhone line at Valley Fair - 10 minutes ago" or "iPhone line was only about 30 minutes in Palo Alto - 1 hour ago." Quickly getting a sense of the crowds before driving all over town is VERY useful.
Google, while definitely not threatened, has no way to provide me that kind of real time info, and until they do, Twitter has an important place in search.