Should I use when or while?

"when" is used because "when" is used with the simple past tense.
"while" is used in the progressive tense. If "while" was to be used in that sentence it would have to be:

While I leave home, my mother gives me some advice I will never forget.

"When" means "after" or "at the time." There is only a few moments in which anything can happen.There is, for example, the amount of time it takes to close a door. The simple past is appropriate.

"While" means "during," and as the other answer says, using the past progressive expresses this. If you use "While I was leaving home," you have, for example, a half hour.