When I was looking for a laptop, I too was considering the Eee PC. Ultimately it seemed the general consensus was "Yeah, it's neat that it's so small, but after awhile dealing with the tiny screen and keyboard sucks." I just ended up going with a full-sized, used laptop off E-bay. It's not so big that it doesn't fit in my backpack or anything like that.
As far as the Eee PC's OS, I think you can get them with Xandros Linux and Windows XP. As far as running Ubuntu, last I looked you were better off with Xubuntu (the lightweight GUI version using XCE instead of GNOME). But that was before a Eee PC with Windows on it even existed so I have no idea if newer ones can handle the OS with more resource requirements. As far as different Linux distros... yeah there are differences but mostly in terms of theming and what sort of software is available, but also in resource requirements. Debian and Ubuntu (and its derivatives) have the most softwae available for them. You wouldn't want to go with SuSE because it's a beast.
I'm not familiar with Xandros or Xubuntu. As far as Xubuntu goes, it's a Ubuntu derivative meaning it should work mostly the same as Ubuntu and have most of the same software available. What I don't know is how desktop environment affects Xubuntu. For instance Ubuntu uses GNOME and Kubuntu used KDE, and they have specific apps. Like on Kubuntu I have Adept instead of Synaptic for package management (that's the software installation tool) like on Ubuntu. Initially, only Konqueror was installed by default when Ubuntu has Firefox (though Firefox has no problems working under KDE). Dolphin instead of Nautilus. etc etc etc
It goes on, but mostly the different software just performs the same function in the end, though some are better than others. I like Adept more than Synaptic due to how it searches packages. Quanta Plus (HTML editor) kicks the shit out of anything that is available for GNOME, though even it doesn't stand up to the mighty Dreamweaver, though you can get that to work via wine (lots of stuff does... like I have Starcraft on my laptop). Basically, I'm not sure how all that plays into Xubuntu. I mean you can get KDE apps working on GNOME, and you can get GNOME apps working on KDE, but you greatly increase your memory footprint because you're basically loading the backend for the whole other display manager, in addition to your native one. I just don't know if there are any "XCE apps," you know?
Anyway, if you are going Linux, I do suggest a Debian-based distro like Ubuntu or Kubuntu. Kubuntu seems more Windows-like to me in terms of how it works. The taskbar layout is very similar, and Ctrl-C copies and Ctrl-V pastes. I like Kubuntu more, but the installed base of Ubuntu is much larger and therefore you'll get better community support with that.
In the end though, any Linux distro could function just fine as a mobile word processor and run Gimp (Photoshop-like app). Personally I love Linux. It took me awhile to sort of get over that hump where I was like "Why doesn't it do x like Windows does?" But once I did that, what I found was a really powerful OS that lets me do pretty much anything I need to do. I can remote into my Windows computers via the RDP/VNC tool. I can SSH into my Linux servers right from the command line. I can listen to my music on my file server no problem.
It's not without its issues though. Before I upgraded to 8.04, I had to install a special tool to get connected to my wireless network because the KNetworkManager bundled with 7.10 was really basic and didn't support shared authentication; it only did open. Also, I had to play Starcraft in a 640x480 window because there were big black bars running vertically when it was in fullscreen, which I initially thought was just that the panel couldn't display that resolution. Then I upgraded to 8.04, and the wireless tool stopped coming back up after a suspend, which I use constantly. I could manually start it, but I couldn't figure out why it wasn't working automatically, everything seemed to be in place and even the Ubuntu forums were flabbergasted. Then I found that KNetworkManager was much improved and I just went back to that and got rid of the Wifi-Radar tool and wi-fi has been fine since. Also after the upgrade, I noticed that now I could play Starcraft full-screen with no bars, but there was no sound. Well a wine update just came out today that fixed that.
So yeah, it's not always smooth sailing and sometimes you want it "just to work." Mostly it does and generally for the things that don't it is a minor issue or something that can be fixed with a little searching on Google.