In general, privately administered IQ & achievement tests may identify whether an individual is gifted, whereas school-administered tests may indicate whether a child is a match for whatever type of gifted programs/services that school may offer.
Essentially true, although I would divide them into individually-administered and group-administered, rather than private and school. Schools can and do administer individually-administered IQ and achievement test. (My understanding is that that is how aeh got her extensive and valuable experience that she shares with us every day.)
