As an adult...I can't think of anything I really don't like that I'm not willing to put up with, or that makes the experience better. I normally wear cloth diapers and plastic pants, so there is noticeable bulk and when they're full they sag...and like others mentioned - they can get really warm. I guess the warmth is the only thing.
As kid...there were a few things I didn't like especially as I got older.
1) When I was wearing diapers in the day and had to use them. It felt really babyish to have to go in a nappy. Especially being asked, "Do you need a change? / Should we change your bum?" or stupid comments like, "Looks like someone has a wet bum!"
2) If I accidentally went #2 it was really itchy and bothersome. I remember the feeling of trying to walk with a full diaper and trying not to 'disturb' the load and making it even worse. And when I got a diaper change, the clean up was longer. I didn't like lying on my back with my butt up having someone clean me up. The wipes were always cold and it felt icky.
3) When I was getting my diaper off in the morning (I was put in them every night) OR when I was wearing them during the day: I didn't like that part of the diaper change just after the pins were taken out and my nappy opened up. It felt cold and I felt exposed, especially when I had to lift my bum up so mom or whoever could take the used diaper out from underneath me.
However, I didn't mind the end part after I'd been cleaned up; when I had to lift up, get powder and rash cream put on. I liked (and still do like) the smell of baby powder. And having a clean and dry diaper put on me. Probably because it meant the worst part of the diaper change was over. I always felt cared for and comfy with a new diaper on.