I would recommend the AT first as sort of "training wheels." Some reasons... One, because you have many more resources to help you organize and prepare. Two, less technical skill needed. Three, logistically easier. Four, the hiking season is longer, which could make your scheduling easier. Five, greater camaraderie and social scene if you're interested. But easy to escape if you're not.
Most importantly, Six: From the bits of the PCT I've seen in person and photos, if you do the PCT first, you'll be absolutely spoiled with the scenery. I imagine it would be very hard to meet that high standard.
But it also depends on where you live, as RyF mentioned. I think if you plan on a Triple Crown, doing them "in order" [AT, PCT, CDT] makes sense in terms of escalating challenge and responsibility. On the other hand, if you have no intention or opportunity to do all three, and you only have one chance, I'd choose the PCT.