Maybe yes and maybe no. I really can't see the difference between multitasking and what I described (very fast app open/close with state preservation) except in a pretty limited number of apps (e.g. pandora). You say I'd notice the difference, but you haven't explained what that difference is. Can you elaborate?
The thing is, you put a new tool into a smart developer's hands and more often than not they innovate and surprise you with something you never thought about. That's why I hate giving a list of examples that I can think of today (not to mention how you always get the fanboys telling you "you don't need it" each time you try to point out something that's missing).
Anyhow... with that caveat, all of these things could happen if multi-tasking / background processing was opened up to developers in the SDK:
1. Music apps (like Internet Radio) that continue to play after an app switch.
2. GPS trackers that continue to log your waypoints after an app switch.
3. Profile apps which automatically switch your phone's configuration, in the background, based on certain events (time of day, location, etc).
4. Phone call filters which help eliminate spam calls / SMS, without interrupting whatever else you were doing at the time.
5. Refresh of app databases (PIM sync, weather sync, etc) in the background, so your apps pretty much always display up-to-date information.
I don't understand Apple's notification scheme yet, so I dunno how many of these scenarios will be covered by it. But, frankly, I don't care. I want developers to have access to this powerful tool to potentially surprise us with things we haven't thought up yet.
A couple other things to consider about your alternative of better state-preserving applications:
1. Dunno if you've ever owned a PalmOS device, but that platform was very, very good when it came to the kind of state-preserving apps you endorse. Yet Palm enthusiast forums were replete with people nashing their teeth over the lack of multi-tasking. The WinMo folks, in particular, trumpeted this as a large advantage for their platform over PalmOS (long before there ever was an iPhone).
2. Apple itself routinely writes apps which take advantage of multitasking (not just state-preservation), yet they deny the technologies to devs through the SDK. If it wasn't all that useful, why do they feel the need to use it themselves?