note down vs write down

To write down, especially hurriedly; to make a note of. 

To write (something) in a simple or condescending style. 

Used other than figuratively or idiomatically: see write, down. 

To produce or set (something) down in writing; to record something. 

To make a downward adjustment in the value of an asset. 

To condemn in writing; to document the faults, offenses, or wrongdoing of.