What are the differences between symmetrical and cambered airfoils?
symmetric wings don't produce any lift at 0° AOA whereas cambered wings do.


Yes, that's correct.

Usually, the most important difference is that a cambered wing is optimised for a positive angle of attack. It produces less drag for the same amount of lift, and can produce more lift before stalling.

Obviously that assumes that the wing is the right way up - a cambered wing will be less efficient and stall earlier when inverted. And a symmetrical airfoil will produce less drag when no lift is required (e.g. a vertical stabiliser in steady flight)