From Marvin in the Discord (#coders-watercooler):
Nano consensus does not verify transactions at all. It’s only purpose is to verify the current state, not the history how we got there.
Light consensus verifies transactions of the last N blocks as well as the state of N blocks back (and thus the state of all blocks in between). It also verifies new transactions to be valid according to the current state.
Full consensus verifies all transactions (old and new transactions) and the state of all blocks