If I have to pick one of the two (AVR vs. PIC10/12/16/18), I would go with AVR, for speed, peripheral uniformity and ease of programming (vectored interrupt controller for example).
However, most of the time, your choice of mcu is influenced by other factors, like existing code base, tools availability, sourcing requirements, corporate policies, etc. The mcus themselves rarely come into play.
If I have to start today on mcu, I wouldn't pick any of them. I would consider PIC24 (if I am also new on coding), or one of the ARM M0/M0+ chips (if I am more experienced in coding).