Roughly, there is no difference. Modafinil is the name of the chemical compound 2-[(diphenylmethyl)sulfinyl]acetamide and is how it is referred to in scientific and medical journals. Even scientists who work for Cephalon refer to the drug as modafinil when they write professionally. Provigil is a brand name and refers to the formulation of the drug modafinil into a medicine for sale (with prescription). Provigil is a trademark under US and international law and is owned by Cephalon. Only Cephalon can use the word Provigil to sell this medicine. Currently, Cephalon also owns the patent rights for modafinil in the United States. They are the only company allowed to make and sell modafinil (except resellers, etc.).
Back in the 19th Century, the German company Bayer invented a new medicine acetylsalicylic acid. They sold it under the name Aspirin. It was very successful and the name “aspirin” eventually became the word for this product. Through common usage, aspirin entered the language Bayer lost any trademark claim they might have. So even after their patent expired and other makers were allowed to make the drug, those new makers were allowed to sell it under the name aspirin. Bayer lost both patent protection and the brand name aspirin.
Nowadays pharmaceutical companies know better. When they develop a drug, they give it two names: the common scientific name (the “generic” name) and the brand name. Cephalon did not invent the name modafinil, but it did invent the word Provigil. (Maybe this comes from “promotes vigilance”?) Even after the patent expires on modafinil, Cephalon will still own the word Provigil for this medicine. Other manufacturers will be able to produce modafinil, but only Cephalon will be allowed to call it Provigil.
Now Cephalon is also developing other brands using modafinil. Armodafinil’s name comes from R-modafinil and is a play on organic chemistry. Modafinil is a chiral molecule, meaning that it has both a left-handed and right-handed configuration. Many drugs are chiral and are sold as racemates (mixture of R- and L- configurations) even though only one of the configurations is active. Nuvigil is another Cephalon packaging of modafinil.