This question does not deal specially with "new" non-euclidian geometries.
Does analytic geometry allow to prove things that could not be proved in the classical euclidian frame?
Conversely, does classical euclidian geometry allow to prove things that cannot be proved using analytic geometry ?
Anything that can be expressed in "classical" geometry can be expressed in analytical terms (lines are functions, intersection points are given by equations). And the other way around. So there is no "real" difference.
However, both viewpoints complement very nicely. Manipulating equations and functions algebraically is easy even for ones well beyond what even Appolonius (of conics fame) did attack, looking at e.g. a diagram helps inmensely at understanding algebraic relations.
But then again, the proofs that some constructions are impossible (like duplicating the cube) aren't possible without looking at the algebraic expression of the problem.