This kinda got me thinking....
I think it's just because with a tattoo or piercings, it's something you REALLY want. You're all jazzed and excited about it. And you're prepared to take the "pain" to get want you really want.
However, with a shot, whether it be at the dr, dentist, or where ever else, it's something you don't particularly want. You're not excited or jazzed about it. And you mentally freak yourself out.