Having worked in various private schools, I'm fairly sure that no distinction is made.
The charges are called "school fees" This covers everything that the school needs to educate the child (teachers, heating the classroom, books, and so on). Before each term, parents receive a bill for the fees. The bill might be itemised, but it is paid in full.
If the child is boarding (sleeping at the school overnight) then there may be additional "boarding fees" (room, heating for the room, dinner, staff to guard the children as they sleep.)
The uniform would normally be bought by the parents, and not part of fees; though some schools may have systems for adding the cost of the uniform to fees. The uniform is not usually made by the school, so the money doesn't go to the school.
We would sometimes talk of "adding something to the bill" for example "Tomorrow we will visit a museum, the cost will be added the bill" (so parents wouldn't pay separately for the museum visit.