Bingley, oh ye of the sheltered life ...
a pump - usually of leather -- (sometimes called a high heel shoe) is a lady's shoe which can be a fully closed, open toed or even sling back shoe, with a heel of 1 1/2 up to 3 1/2 inches.
A stilleto heel is at least four (4) inches and can be more. Stilleto heels are found on pumps.
So, all stilleto heeled shoes are pumps but not all pumps are stilleto heeled.
Stilleto pumps are sometimes connencted to shoe fetishism.