The terminology is wrong: the polynomial curve fit is still linear regression. The "linear" in "linear regression" does not mean that the resulting function is linear, it means that the resulting function is a linear combination of basis functions. For a polynomial, the basis functions are 1, x, x^2, ..., x^n.

The exponential fit, however, is non-linear.

From Wikipedia (http://en.wikipedia.org/wiki/Polynomial_regression)

> In statistics, polynomial regression is a form of linear regression...