Trees (in particular, homogeneous) are discrete analogues of Cartan-Hadamard manifolds (in particular, of simply connected manifolds of constant negative curvature). Although dealing with trees is much easier technically, they were considered much later: function theory, harmonic analysis, automorphism groups, random walks vs Brownian motion, representation theory etc. One has to admit that mostly (not always, though) it was done by direct translation (sometimes almost verbatim) from continuous into discrete language.

Another example is provided by the discrete potential theory (sometimes interpreted as the theory of resistive electrical networks). Here, once again, in spite of being much more elementary it was developed significantly later than the continuous theory. I would say that in the latter case the discrete theory is more independent than in the case of geometry on trees.

Yet another example (where the discrete part is much more original) is buildings vs Riemannian symmetric spaces.