Hiked parts of both. Read extensively. Talked with many who have completed both.
PCT: Almost all accessible by pack animals. 500 more miles. Grades are easier. Trail is much, much easier. Less elevation gain than the 2100 mile AT. Mostly out in the open. Better scenery overall. Supplying is harder. Level of fitness required to complete it about equivalent to AT. Water is hard on the PCT.
AT: Here on the east coast, where I live, we don't have majestic mountain ranges. To compensate for this we blaze our trails up and over every mountain in sight. The elevation gain is much more than on the PCT. Parts of the AT can be brutal. You could do 20-30 per day on the PCT whereas 20-30 on the AT would be a significant challenge. Water and supply is much easier on the AT. Springs are everywhere and it is very easy to manage. Often referred to as a "Long Green Tunnel" because of how closed in everything is. The elevations are lower so you don't often break out of the tree line. Bugs, bugs bugs in the summer. Cowboy camping is harder to do on the AT, but quite feasible in many parts of the PCT. Also the trail terrain on the AT is much harder. Rocky/uneven, that is implied by my other comment on the PCT, but to be clear, it is rocky and in some parts downright brutal.
The PCT is logistically more difficult than the AT. The AT is physically more difficult than the PCT. Both are long distance hikes where, once initial physical issue have been conquered, become more of a mental challenge than physical challenge.
The PCT seems to be more hostile to feet than the AT. Lots of hikers have a difficult time figuring out the right combination of miles, shoes, socks, and so on during the Southern California part of the PCT. Blisters are very common. The AT's ups and downs can be more hazardous to ankles and knees. Hiking poles are common and useful on both.
Resupply options on the PCT are less frequent than the AT leading to longer sections between resupply than on the AT. There are only three or four shelters on the PCT so camping locations must be thought out in advance (i.e. consulting maps to identify level ground) or discovered en-route. You have to be prepared to dry-camp as stopping for the day at the last water source will cut into your daily miles too much.
Weather can be challenging on either trail but the 'season' for hiking is much longer on the AT than the PCT. PCT hikers can't spent lots of time in town if they want to finish before the snow arrives in Washington.
Can you recommend any resources for someone that wants to hike the PCT? I'd like to do it in the next couple years, but the logistics have intimidated me. I've hiked the JMT twice, and doing it again this year. I'm not really even sure where to start with doing something like the PCT.