What happens when you turn 13 and 15 in The Giver?

There are no special privileges or ceremonies awarded to people who turn 13 or 15 in the book The Giver.

At each age up to 12 the turning of each new year is celebrated by a group of ceremonies that signify the aging of each child. However, 12 is the last year that ages are celebrated or even recognized. Jonas' mother says in chapter two that:

"After Twelve, age isn't important. Most of us even lose track of how old we are as time passes, though the information is in the Hall of Open Records..."

Basically, the preparation for adult life takes over the childhood of people within Jonas' society. When children turn 12, they receive the career assignments that they will follow for the rest of their lives. After that point their age becomes a useless construct that does not affect them, as long as they are able to continue to carry out their work.