What’s the difference? The process.
Cold Foil
A UV-curable adhesive (ie. a type of glue that is dried by UV light) is printed on the paper in the shape of the design that is to be foiled. Then the foil is pressed onto the paper, and is stripped away where no adhesive was printed. After a quick run under the UV light, the product is finished. (See a simple diagram of the whole process on Wikipedia.)
Hot Foil
A die, or stamp, of the design to be foiled is mounted on the stamping machine above the paper and is then heated. The sheet of foil runs between them so that when pressure is applied from the die onto the paper, the foil is fixed to its surface. Because pressure is a main ingredient to hot foiling, the foil has dimension and a distinct feel.