The core difference is what is being ratcheted.
Signal's Double Ratchet evolves the keys for a static algorithm like AES.
In Ariadne, the ratchet evolves the algorithm itself. The path through the Labyrinth, the sequence of cryptographic permutations, is a function of the CVM's state. The state ratchets, so the algorithm ratchets.
Signal gives you a new key for the same lock. We use a new key to build a new lock for every message, and the shape of that lock depends on the history of every one opened before it. This architectural difference is what allows the CVM to be reconfigured for other tasks like VDFs or cooperative proofs-of-work, where the "program" itself must be dynamic.
Looking at it naively - deriving a new key sounds similar to picking a new function within a family of possible functions?