Hmmm. An entire OS architecture built around a weather app. Sounds like overkill to me.
I believe that was just an example. Ok, I'll give you some useful ones that I actually like:
Stock App- No need to run anything. In under a second I see how the DOW and my top 5 stocks are doing. Contrast to having to run the app, let it load, let it download, then display. Extremely useful for me.
Word of the Day- New word of the day pops up. If it's an intriguing one, I click in and learn it.
MagikMail- hooks into my YouMail system. Tile flips to show me who the last person who left a message is. Nice for when I miss a call and want to know if I need to check it asap or not.
News- BBC News Beta, in particular. Get to see the hot / breaking news as a picture + text just on the tile. Click in to see the whole thing.
RSS- Running updates of number of unread stories in a given feed.
People / People Groups- (I use groups). Out of the 10 people I actually want to know about, it pops up the latest post or thumbnail picture they shared.
British Air - don't think this one is out yet, but you can see it demoing mango features. Updates itself as to a particular flight with delays, info, etc.
I think you can see some of the benefits of live tiles. Not useful for every app, definitely, but very nice for some scenarios.