The reason why silver dissolves (or water is oxidized) is well known. The reason why platinum does not dissolve is more elusive. Thanks to Maurice's comment for pointing this out.
Let's assume your solution is 1M in AgNO3 and 1M in acid ($H^+$). The potential required to oxidize silver is 0.80 V; the voltage required to split water (oxidize it) is 1.23 V, but there are complications of overvoltage, mostly with hydrogen, but also with oxygen. To sum up: it is easier to grab an electron from silver than to pump another hydrogen ion into solution. AHA! How about reducing the acid concentration, or not adding any at all? (Of course, the reason for adding the acid was to make the silver dissolve easier.) If the pH is increased so that hydrogen ion concentration is reduced to give a lower oxidation potential than silver, then water will be oxidized, hydrogen ion will be produced, and eventually, silver will become more easily oxidized. This electrolysis wil go thru a phase where hydrogen and silver are codeposited, probably making a porous, weak deposit of silver.
For platinum to yield 2 electrons requires 1.188 V. Although this is less than required for oxidizing water, overvoltage effects due to oxygen and hydrogen are apparently important. Although platinum can be electroplated with a platinum anode, apparently the process fails to work well; older methods of platinum plating were kept refreshed with additions of platinum salts rather than relying on dissolution of the anode (Ref 1). More modern methods use a pulsed electropotential, but their success seems to be from a solving difficulties at the cathode rather than getting the anode to dissolve (Ref 2). They deposit a monolayer of platinum attached to a monolayer of hydrogen, then oxidize off the hydrogen and repeat. The electrode potentials strongly suggest that increasing the cell potential, if you had a platinum anode, would result in dissolving platinum metal, but the catalytic reactivity of platinum with respect to H2O2 suggests that maybe the oxidation reaction occurs in the water layer, and the platinum acts merely as an electronic conductor.
Ref 1. https://www.technology.matthey.com/article/25/1/32-41/
Ref 2. https://www.chemistryworld.com/news/platinum-plating-at-the-flick-of-a-switch-/5720.article