Woofers are designed to operate at frequencies of 100hz and above. They tend to have a little voice that comes thru. They are generally used in home audio speakers cuz they can.produce a little bumps along with a middle flat sound which blends in with maybe a 4" midwoofer and a 2" tweeter.
Subwoofers operate on lower frequencies from 100 hz and down to 20 hz. Many of the sub frequencies are inaudible but can be felt by your body or even travel long distances. Most subwoofers are set up that no voice sound comes thru. If you want a solid more refined bass sound then your gonna go with subwoofers. Most car audio subs are 10" and above. Tho u can get 8" that produce in sub frequencies most 8",6",5" etc...are woofers or midwoofers.
Alot of people like 12" subs because they are versitle to different types of music. They can reproduce rock music and rap music where as 10" fair better with rock music or punchy type bass. Tho 10" can reproduce low frequencies given the right brand and box design