I wonder what's the difference between these two verbs given that the dictionaries translate it the same way?
Piss would be considered more vulgar, pee would be considered more mild and informal.
It's common to ask a young child, "Do you need to go pee?" I wouldn't ask a young child, "Do you need to take a piss?"
The words also have differences in how they are used, as those questions show. You can go pee, and you can take a piss. (Other variants may apply, with differences in how standard they might be. I wouldn't say that take a pee or go piss are unacceptable, but they don't seem as common.)
Piss can also be used a slang verb. To get pissed is to get very angry. (For example: Joe was pissed about his traffic ticket.)
In my (American) experience:
"Piss" is a word used by men, and by boys who were raised by men.
"Pee" is a word used by women, by girls, and by boys who were raised by women.
"Urinate" is a word used by doctors, nurses, and other medical professionals.
Both terms are informal and considered vulgar slang. Piss is somewhat onomatopoeic and comes from the Old French "pisser" (PIS - ay) meaning to urinate.
"Pee" is just the first letter of "piss" and meant to soften it and make it more polite. This may be a reason some associate the use with children and the female gender.