At any fixed-price restaurant (the type where guests are charged per person instead of by item they order) a 10 year old pays the higher price (Disney doesn't use the term "adult" or "child" for tickets or dining plans, they say "guests age 3-9" and "guests age 10 and older").
However, at a la carte restaurants, for the most part it appears they are able to order from the children's menu at children's prices if that is what they prefer.
It does make a difference whether you are talking about fixed price places, like buffets, family style all you can eat restaurants, or fixed menu restaurants, and whether you are discussing a la carte restaurants. Unfortunately for a lot of families with older kids, all character meals are fixed price, and the
Disney Dining Plan requires the higher price for children age 10 and over.
They pay the higher price for the tickets because by the age of 10, most children are physically able to participate in all the available rides/shows if they choose to. Disney didn't just set that age arbitrarily. As for the restaurants, I think for whatever reason they wanted to match that up with the tickets, possibly to assist in booking of package reservations including tickets and dining. The restaurants used to be 12 years old for the higher price, not 10.
They still use the designation "junior" for children aged 10-17, but only regarding resort room reservations. A "junior" is a registered resort guest who is too old for the child designation, but not old enough to kick in the extra adult charge on the resort rooms.