Their idea of a lightweight computer starts at $1800.
Well, no. Apple's idea of a cheap, lightweight computer costs $300 to $500, fits in your pocket, and (optionally) doubles as a cellphone.
(If you don't agree with this interpretation of Apple's product line, wait a couple months after the official third-party iPhone apps arrive and consider it again...)
You can't type on the iPhone very quickly, of course. It won't run emacs or vi very well. It probably won't run a web development stack, or a decent Java compiler. So if what you really want is something like an EEE you should buy an EEE. There's no reason why Apple's idea of a cheap, lightweight computer should match yours...
Whether or not the iPhone is in the EEE's market segment depends on what you think that segment is. I'd say that it's squarely in the "super-light super-cheap portable box for checking mail and web" segment. If you're talking about the "light, cheap portable box for fixing your website via SSH from a coffeehouse in the Florida Keys" segment, it's a lot less clear.
I'm sure there is some substitution between the categories, but I guess I still think of them as pretty separate.
Pesonally, I have a phone that runs both ssh and vi (a Samsung i760 Windows Mobile 6 phone with a touchscreen and slideout keyboard), and I'm still interested in getting an Eee.
These appliances are connected devices, which means that all they need to be are small wireless dumb terminals. Increases in bandwidth and improvements in distributed computing (cloud computing to be buzzword compliant) will make this even more so.
Meanwhile, Linux is eating their lunch in this segment.