A "proof test" is a test to failure.
A hydrotest is a method of testing, i.e. using water or hydraulic fluid rather than a gas.
Typically, hydrotests are nondestructive tests carried out to a small multiple of the maximum allowable pressure that the object is designed to withstand. However, a hydrostatic test which is carried out until the part fails is called a proof test, and is done routinely to demonstrate the pressure rating of assemblies such as hoses where you can't determine the MAWP reliably by calculations alone.