They are not exactly the same. "I'm disappointed in you," communicates how you feel about what John did. "You let me down", communicates more matter-of-factly what John did, although one might expect it to make you feel disappointed. Also, one can be/feel disappointed over things beyond anyone's control. "I'm disappointed that the game was canceled because of rain." "I'm disappointed that I didn't win the lottery." To be let down, on the other hand, some one specific generally lets one down by failing to do what they said they would do or what was expected of them.

There are more differences (apart from one being an adjective and the other a phrasal verb). "Disappointed" has a more formal tone, and "let down" is suitable for regular, everyday speech. Also, like many other phrasal verbs, "let down" has other meanings. Rapunzel famously let down her hair, to mean she allowed her hair to fall out the tower window and to the ground below. No unhappy bunnies there. :) Using it as a rope, she could also let other people down (ie. help them descend), and even let herself down. "Let (someone) down" can also mean allow someone to descend, if that person has been trapped in a high place and depends on someone's decision for their freedom. If the object is inflatable, then you let down a car tyre (for example) to mean you open the valve and allow air to slowly escape.

Yes they are the same.