Premiere is primarily a video editor. transition from clip A to clip B, etc.
After Effects is primarily a compositor, for like advanced titles, flying around stills, moving shit around in a 2d or pseudo-3d space, animation of layered PSDs or AI files or logos or whatever.
they share many attributes to greater and lesser degrees. you could edit video in AE and it wouldn't be the end of the world. likewise, you can do titles in premiere, though i can't speak much on that. I never open Premiere, and i think you can tell from my descriptions that most of my experience lies in AE-world.
they share a lot of effects and presets, last i looked.
(Edit... I should have refreshed the page... Tom spelled it out perfectly.)
They're not really interchangeable though. After Effects is kind of clunky for editing video. Adobe loves to point out that it's not a non-linear editor at all... usually when users complain about the way previews work. Anyway, if you want to take multiple sources and edit them down to one video, Premiere is the one for that. For titles, effects, or just plain animating stuff... After Effects all the way. It's kind of like Illustrator vs. Photoshop... they both have their job, but they aren't quite the same. I beta tested Premiere .8 in our college lab a million years ago and used it until version 4.5... Unless they really messed it up, it's pretty easy to learn. After Effects is surprisingly intuitive as well. Get both.
sit at barnes and noble and read $40 books. ho ho ho.
yes, i would certainly not edit, say, a sitcom in After Effects. But I have edited together smaller like 5-minute stuff, but that doesn't mean it's the better choice, just the interface i'm most comfortable with.
an example workflow might be to stitch together footage of, say, an interview in Premiere, then switch to AE to make a posh little title animation of the person's name for the lower left corner, render w/ an alpha channel, and bring that new clip back into Premiere to plop on top of the edited video. I know the Creative Suites are integrating everything together more, so maybe even the rendering step isn't necessary anymore. that's beyond my skillset. But, for another instance, all that Arrested Development title stuff I'm doing for that documentary, I am delivering via email, just sending the dude quicktime movs to drop into final cut pro or whatever.
oohh, "Design Rhythmic Motion Typography in After Effects"... i'm not one to get jaded on trends cos I usually don't notice them, but this stuff was beaten like a dead horse in aught-eight. heh. (I made one for Tesco and I still feel a little dirty)
What I tend to do is edit the video in Premiere, and do basic effects (most of them you know from Photoshop filters) in there too... Whenever it comes to trickier stuff (like adjusting just a part of the video - say you want to shade a single walking figure in a bigger picture), i'll swop to after effects just for that bit of film... and bring it back in premiere afterwards.