Difference between Non linear regression vs Polynomial regression
the difference is probably easily seen with an example. Linear regression assumes a form $$f(x, \beta) = \beta_0 + \beta_1 x_1 + \cdots + \beta_n x_n$$ with some covariates $x_i$ and some parameters to estimate $\beta_i$. An example of non-linear regression would be something like $$f(x,\beta) = \frac{\beta_1 x_1 + \beta_2 x_2}{\beta_3 x_3 + \cdots + \beta_n x_n}.$$ Essentially you are assuming your model to be of a nonlinear form. Polynomial regression on the other hand is a fixed type of regression where the model follows a fixed form $$f(x, \beta) = \beta_0 + \beta_1 x + \beta_2 x^2 + \cdots + \beta_n x^n$$ which is a nonlinear function, however it is still linear in the parameters $\beta$ you are trying to estimate.

That is to say, 

Polynomial regression is non-linear in the way that $x$ is not linearly correlated with $f(x, \beta)$; the equation itself is still linear. 

In the other hand, non-linear regression is both non-linear in equation and $x$ not linearly correlated with $f(x, \beta)$.