Successfully complete the game to unlock the Madhouse difficulty. The following things change in Madhouse mode:

Health does not regenerate.
Enemies and Stalkers appear more frequently, and in new locations.
Additional Antique Coins (33 in total) and more locked item bird cages.
New items; Cassette Taps must be collected and used for manual saves.
Fewer items; many items from the Easy and Normal difficulty have been removed.
Few auto-saves. You almost always restart from your last manual Save Room file.