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.