Why do buoyantly rising mantle minerals melt, yet buoyantly rising atmospheric waters condense?
There is no comparing the chemical composition of "atmosphere" with that of mantle rocks and the consequences of rising in each have almost nothing to do with one another. The cooling of rising atmosphere is a fairly simple thermodynamic process requiring neither molecular change of ingredients, chemical release of water, nor thermally impervious boundaries.

By contrast, rising mantle materials will undergo different melting processes depending on their water content and chemistry. In the simplest case, like a mid-ocean hot-spot, unpolluted by melt fluxes like water and carbonates, rising mantle rock undergoes "decompression melting" in which 2 things happen:

1) The pressure needed to keep rising mantle material solid is released and the material becomes liquid (like that crudely drawn by Michael, above). Remember, given enough pressure, you can heat up ANYTHING and keep it solid...even hydrogen, like that believed to exist in the earth's core.

2) Constituent minerals undergo phase (mineralogical) changes on their way up to the base of the upper crust. Remember that mantle materials come to the earth's surface as basalts which are mineralogically different from the olivene-rich peridotites of the mantle. Despite being chemically similar, deep mantle rocks contain minerals that are crystallographically denser than those found basalts. As they are brought up, these minerals undergo a phase change that alters their crystalline structure; they become different minerals, lighter and more buoyant, further reducing the density of the rising melt.

So, there it is (in a nutshell): reducing pressure on hot solid material allows it to become liquid and also invokes phase changes in minerals that make the overall material less dense.