Can't run two separated DC Motors
Connect the Nano ground to the driver board ground.  Since they do not share a common ground, they can have odd behavior as their ground planes float.

You need only connect the grounds, the USB can still provide +5 to the Nano, and the +9 can still provide power to the driver.  The only energy that will flow through the black wire connecting the grounds is the energy that flows between pins 2,3,4,5 and the driver's AIA, AIB, BIA, BIB, for the logic signals, which I'd bet is in the sub-uA range.