Tape the purple wire off where it cannot contact any surfaces. It is not needed if you wish to connect back the way it was with the old motor HOWEVER only do so if that cap matches your new motor's microfarad rating or it will get hot and quit take a dump and generally piss you off.
Connect brown and white striped wire to C on the capacitor as it is COMMON and the same as purple.
The solid brown wire goes to F for fan on the dual cap. Connect black to the contactor.
As far as do you need a new capacitor? HELL YEAH. My opinion. Would you use your old spark plugs in a brand new vehicle? The person that sold you the motor should have insisted on selling you a matching cap and shown you how to connect it to keep poop down. Just sayin