It was such a simple question, but you guys gave such complicated answers:(
It is really not that hard to answer: in both cases, we can model the reflection as light (an electromagnetic wave) passing through a medium with changing index of refraction. In both cases, we can analyze it as waves or rays, but it is better to analyze both as waves. In both cases, waves are reflected when the index of refraction changes (as Rennie already pointed out). In that sense, it is the same mechanism.
Now true, technically, what happens in the ionosphere is a bit more complicated, with radio waves being bent gradually back to earth rather than reflected at a sharply defined boundary with a discontinuity in index of refraction, but that is ignored when actual users of such radio communication (such as radio amateurs) work in terms of the (virtual) height of the relevant layer of the ionosphere (layers D through F2). See http://www.electronics-radio.com/articles/radio/basic_radio/propagation/ionospheric-hf-propagation.php for a simplified explanation of ionospheric reflection, http://www.qsl.net/zl1bpu/IONO/iono101.htm for more details.
Oh, there is one other aspect in which they are different: since ice and air are both dielectrics with ice having the higher index, the reflection is phase inverting. We don't usually care, because the phase is lost in the diffuse reflections anyway.
But since air in the ionosphere is conducting, we have to model it as a reflection where one medium is dielectric and the other conducting, so the treatment of phase is different. If it were a good conductor, the reflection would be phase preserving; but since the ionosphere is rather weakly conducting, the full electromagnetic wave treatment of this case is rather difficult, and is not found in elementary E&M texts: one has to resort to odd or old sources like Frankel to see it fully analyzed. Yet as the QSL site reference above mentions, one can at times hear the effects of phase distortion in HF signals reflected off the ionosphere. So the difference is relevant.
Come to think of it, my own answer turned out to be more complicated than I expected when staring it, too;) But I hope it has answered your question more fully.