Why is the OO-H dissociation is unlikely for the Caro's Acid?
One possibility is the formation of an intra-"molecular" hydrogen bond between the peroxy-bonded hydrogen and a third oxygen atom.  The structure shown in Wikipedia for the acid suggests this interaction.  The involved hydrogen is curled up into the five-membered ring formed by such a hydrogen bond, while the other hydrogen atom is freely exposed and ready to launch.  Potassium peroxymonosulfate corroborates shows only the non-peroxo hydrogen dissociated.

Hydrogen peroxide does not form this sort of intramolecular hydrogen bond, and as we expect from induction considerations the proton dissociates from this molecule more readily than from water.