More generally, if I want to know what now is happening about something, twitter can usually deliver that.
There, you have used Twitter in the fashion of the "traditional" web, you just went to a RTW source thats all.
My point is i generally don't have any use for what "now" is happening, anything that is of importance generally has a longer time span and relevance to me and i don't "need" to know it straight away.
We're not used to having down-to-the-second real time search yet, so it's harder to come up with use cases until we really become comfortable with these new systems. Most use cases we do see, like Trending Topics on Twitter, are discovery queries that will eventually be covered by Google's spidering, but there are whole categories worth of search cases in the long tail that won't be covered by just discovery.
Other uses might be traffic information, black-friday sales, tornado tracking/response, stock trading, organizing and coordinating civil protest, and rapid response for brand managers.
Surprisingly, the only exception to that is probably FreeNode IRC , certain channels at least.
Yes, there are certain bloggers i want to read, certain discussions on HN that i want to participate in but it takes time for me to assimilate and understand what is going on, the RTW doesn't help me there.