I found this nice simple definition on Usenet that I like:
With a polynomial, it is the base number that varies. The exponent is given.
Ex. x\^2 (x is the base number, 2 is the exponent.)
With an exponential, the exponent varies. The base number is given.
Ex. 2\^x (2 is the base number, x is the exponent.)
Thus, as x increases, the exponential function grows much faster than the polynomial.