Non-Synthetic - "Natural" Weed Killers: 

There are 3 primary "natural" products on the market: Corn gluten, acetic acid/clove oil combination (A 4th product was recently introduced with a "food-grade" 20.0% Acetic Acid), and fatty acid. Nature’s Avenger’s patented d-limonene (citrus oil) formula is a new fast acting and effective alternative - we’ll get into that later

Corn Gluten: Used as a pre-emergent 
weed killer. In other words, it is applied BEFORE weeds emerge. Corn gluten inhibits weed seed germination. It is now considered an organic product.

Acetic Acid/Clove Oil Combination: (acetic acid is vinegar) Used as a 
post-emergent herbicide. It is applied AFTER weed seedlings emerge. 

Now, it gets a little complicated! Natural vinegar contains only 6% acetic acid. New products (for example, Burnout II) are coming out with a much more concentrated acetic acid (some as high as 30%). There is NO way to naturally obtain acetic acid at these high rates; they are all produced synthetically (it’s a simple, i.e. cheap, chemical process). Since these products are made with synthetic acetic acid they will never be considered "organic".  If these products are said or listed to be organic, they are doing so incorrectly. Products containing such a high concentration of acidic acid can also be dangerous. Make sure you check the MSDS sheet for handling of these products and have the proper safety gear.

For example: the company's MSDS on Burnout reads the following:

Health Hazards: Contains acetic acid and is flammable and extremely corrosive. 
Contact with this product will result in severe eye irritation and possible permanent damage. Contact with this product will cause severe skin irritation and/or chemical burns. Breathing vapors will cause significant respiratory irritation, and pulmonary edema if prolonged. Ingestion of this product could cause burns and destroy tissue in the mouth, throat, and digestive tract.

Also, products with clove oil tend to have a strong, rancid odor. NAO has a fresh citrus aroma.

"Food Grade" 200 Grain Vinegar or 20% Acetic Acid: Again, this is used as a post-emergent herbicide. As stated earlier, natural vinegar contains only 6% acetic acid. When you get into a high concentration of acetic acid, danger looms for the homeowner/user. Homeowners are not trained in safe handling and storage of concentrated chemicals. Acetic acid concentrations over 11% can cause burns upon skin contact. Eye contact can result in severe burns and permanent corneal injury. Products with high acidic acid concentrations registered through the EPA for commercial use, all have restricted entry intervals of 48 hours and list personal protection equipment to be used by the applicator. Because consumers are used to thinking of vinegar as something you can safely use on your salad and eat, they are generally unaware of the potential dangers of higher concentrations. 

An example is a copy of the MSDS sheet warnings on Weed Pharm®, who claims "food grade" 20% Acetic Acid:

This product contains hazardous materials as defined by the OSHA hazard communication. Vapors irritating to the eyes and respiratory tract. Avoid inhalation. Liquid may cause eye burns and permanent damage to cornea, possibly leading to blindness. Causes skin irritation.EYE: Extremely irritating to the eyes. If not removed promptly, will injure eye tissue, which may result in permanent damage, including blindness SKIN: Contact causes skin irritation and possible burns INHALATION: Irritating to the nose, throat and respiratory tract. Avoid breathing of vapors or mists INGESTION: Can irritate or burn mouth, throat and stomach if swallowed.

Please be carefuly when using these types of products.

Fatty Acid: (essentially soap) Used as a post-emergent herbicide. It is applied AFTER weed seedlings emerge. The fatty acid products (soap), are practically useless. But in the past couple of years, a short chain fatty acid (C9 or 9 carbons) has been discovered that is very active. It is available only synthetically (i.e. not natural). The C9 fatty acid is called pelargonic acid and is sold under the trade name Scythe Herbicide. Scythe is expensive and is NOTorganic.