How are they different? and why would would the second regression be more appropriate?
$ Y = A_1(x_1)^2 + e $
$ {\sqrt Y} = A_1x_1 + e $
My thoughts: Square root of a function makes the function more linear, so the second regression would be preferred due to this reason.
On
Which regression you would prefer depends on what you think is the distribution of $e$. Imagine that the data is generated (i.e., the "true model" is) with $e$ as Gaussian in your second equation. Then if you changed to the first form, then the $e$ term would not longer have the Gaussian distribution and OLS would not be appropriate.
+1, btw on @Claude Leibovici's excellent remark.
A model is said linear when it is linear with respect to its parameters; this does not have anything to do with the variables.
For example, $$y=a+ b x^\pi+c e^{\sqrt{x^3+9}}+d \sin(8x)$$ is a multilinear model.