Trust region is more robust if you have strong non-linearity. This effect is "amplified" depends also how far the starting point from the true solution.
The downside is it project the "Hessian" on a small subspace (2nd dimension), so it will not converge rapidly if the function is convex but with large difference of the amplitude of the curvatures. But it can deal with with local negative curvatures.
Levenberg-Marquardt requires to evaluate the Jacobian, which can only effectively computed in small/middle scaled problem. As I said, it's approximate the Hessian with J'*J, so it's more accurate for problem with medium non-linearity.
In short: Trust region more robust, used for large scale, strong non-linearity. (typo EDIT)
Levenberg-Marquardt , less robust, used for medium scale, medium linearity, or the first guess is well estimated.