I've never done the AT, but I've heard lots about it from other hikers. The PCT is a more modern trail, developed with horses in mind as well as hikers, with a real trail "standard" of width and grade, etc. In general it's wide, open, well-marked, and well-graded. It's easy to do big miles on every day (if there isn't snow on it!).
From what I know, the AT is a little older, with lots of PUDs (pointless ups and downs), just a "tougher" trail to do big miles on. Roots, rocks, ups and downs. It sounds like 20 miles is a real long day on the AT sometimes, while 30 is a long day on the PCT.
Other big differences are long stretches without water on the PCT, both potentially in southern California (in the chaparral, coastal sage, and real "desert" areas), some in Northern, and defniitely in Oregon too. However, you don't need water drops of your own or need to rely on those done by trail angels - it's not difficult to carry enough water to get from source to source, and I think it's important to not rely on the water caches - too many bad stories of people relying on them, to find them empty when they arrive. Carry what you need!
Much more relief on the PCT due to the bigger mountains. Also, from what I have heard, a lot more scenery and views. I had a friend say that you see more scenery on the PCT in a week than you see in a month on the AT. They don't call it "the long green tunnel" for nothing. Weather wise they can both be tough, no doubt. The PCT can have rain, snow, super hot and dry, and everything in between - but at least no humidity! In general, you won't see almost any rain all summer on the PCT, unless it's afternoon thunderstorms - as you head towards fall in the Cascades, you can get rained on (or snowed on) a lot... I know we did in 2004.
Another one is that you will be much further between town areas on the PCT in general, especially as you get to OR and WA. There are maybe two shelters in the whole 2700 miles. It seems like more of a "wilderness" experience to my knowledge. As you get further north, the number of hikers can get low and spread out... I know that I hiked for days by myself and sometimes didn't see anyone else the whole day, or just an elk-hunter. Lots more people on the AT, and from what I have heard a lot more people section hiking, partying, kicking it, jumping on and off, etc.
I somewhat agree with Casey about the difference of the JMT section and rest of the PCT (if you don't know, these are the same trail for almost all the JMT's length). I would say that there are plenty of sections on the rest of the PCT that are similar - there are plenty of mountains to go up and back down, both in SoCal and especially in the Cascades. And the section north of Tuloumne Meadows is definitely up and down, hopping canyons. The JMT section is just more dramatic, with above timberline passes and then down into canyons, and then repeat, and then repeat, for 10 days or so. It's amazingly pretty, of course.
towards muir pass on the PCT/JMT: