"Sit down" just takes its literal meaning - as in "Please sit down and not remain standing." "Sit well with" means "be harmonious with", and the Oxford Dictionaries site gives this example: "His shyness doesn’t sit easily with Hollywood tradition." It can apply to matters of taste: "Much as I admire eclecticism, this Chinese Ming chair really does not sit well with your Louis XIV secretaire." It has nothing to do with sitting down, but the two pieces are just aesthetically incompatible.

Sit well with someone - to be acceptable to someone. For example: " 1. Your explanation of your absence doesn't sit well with the president. " 2. " The whole affair didn't sit well with the manager. "