I am working with a numerical application where some known techniques solve $\Delta u=\rho$ for $\rho(x,y)\geq0$ defined on a planar rectangle (given as the density of a large pointset). Since the application is related to transportation problems, I suspect that solving Monge-Ampere $\det(D^2 u)=\rho$ (with the same boundary conditions) may work better. Hence, the following questions. 


1. What are the most striking differences between the two PDEs ? It would be useful to have examples of $\rho$ leading to very different solutions, and to identify general properties of individual solutions that differ between the two equations. For example, do solutions of Poisson's equation with nonzero $\rho$ minimize some useful functionals ?
2. Are the *best* numerical techniques for these equations essentially the same ? The answers may differ if *best* is interpreted in terms of speed/convergence, robustness, ease of programming, support in Matlab, or availability of open-source solvers.
3. Should I expect the (nonlinear) Monge-Ampere equation to be more difficult to solve (numerically) than the Poisson's equation with the same $\rho$ and the same boundary conditions ? Is it possible/practical to solve Monge-Ampere by somehow solving/resolving Poisson's equation ?
4. What are good references on these topics ?


Just in case, my $\rho$ can be heavily concentrated in small regions, often vanishes in other regions, and may experience sharp steps-like discontinuities. But this can be papered over by Gaussian smoothing (at some cost).



Differences between the Poisson's and elliptic Monge-Ampere equations?
I agree with Deane Yang's comment and will try to expand on it in this answer.


First, the standard formulation of the Monge-Ampere equation is $ \det(D^2u)=\rho $.


Second, the Monge-Ampere equation and the Poisson equation are completely different, and the numerical techniques for the two problems are very different. The Monge-Ampere equation is fully nonlinear and will in general be much more difficult to solve numerically. 


Finally, a recent paper on numerical methods for Monge-Ampere is S.C. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung, C0 penalty methods for the fully nonlinear Monge-Ampere equation, Mathematics of Computation, 80:1979-1995, 2011. The references inside should give you a starting point for your questions.


