Charcoal briquettes were made popular by Henry Ford, when he was looking for a way to use the leftover scraps of wood and sawdust from the manufacture of his Model T automobiles. He and his brother-in-law E.G. Kingsford, who brokered the site selection for the original charcoal plant, started the Kingsford Charcoal Company, and until well into the 1950's one could buy charcoal at Ford dealerships throughout America.
Lumpwood charcoal on the other hand has been around for centuries, as is simply partially burnt, odd-sized pieces of hardwood such as oak, hickory, maple or mesquite. Lumpwood charcoal may be marketed specifically as "mesquite" or "oak", or simply as "lumpwood" charcoal, which is usually a mixture of several hardwoods.
The difference between the two is vast. Briquettes are held together with various binders, and if not allowed to properly pre-heat, can impart off flavors in food. On the other hand, it is consistent, predictable, longer burning, and can produce excellent results.
Lumpwood charcoal has been referred to as like cooking with charcoal 'on steroids' It burns much hotter and faster, and of course is all natural. Barbecuing (i.e. low and slow) with lumpwood means more often refueling, and a more watchful eye.
Lumpwood charcoal is nearly always my preference, but I will use briquettes occasionally for a long 12 hr. smoke. Both will produce superior results to your gas grill.
Look for lumpwood charcoal in 40 lb. bags in your local big box or superstore for around $15, as opposed to paying $8 for an 8 lb. bag at the grocery store. Happy grilling!