A couple off the top of my head:
When I only have a couple of minutes (elevator, bathroom, waiting in line) and want to check my RSS reader it's great that it's already up to date and I don't need to spend one or more of those minutes updating all of my feeds.
When I'm listening to pandora and I get a phone call it's nice to not have to go back to my home screen, click pandora and wait for it to start up again before I can listen to music after ending the call.
When I'm logged in to IM I don't necessarily want to be looking at it the whole time.
And many more. Push can 'solve' some of these, but requires lots of infrastructure and reliability from your app vendor, and isn't necessary if your apps are allowed to run in the background. Sure, it requires your app writer to not suck at writing a persistent task, but at least it's an *option*. If it's a problem for you, and battery life is at a premium, don't install the apps, or shut them down when you're done.
When I use my phone I'm never doing it for long periods of time, only minute or two intervals, and if I have to start and update an app in that time it's not nearly as useful.
My BlackBerry 9700 easily survives two days on a charge.