Everything written here about Whitney v. Rainier is very correct. I would just say from my perspective (comparing classic routes on both mountains) that Whitney is just hiking, Rainier, on the other hand, is mountain climbing. For both, however, one has to be in a very good shape and know how to control the paste to avoid a high altitude sickness.
Some time ago I tried (successfully) to climb each mountain in one day (never tried Rainier). But than I realized that I was missing my enjoyment, especially views. I changed my goal to sleeping on the top of a big mountain I climb, at least to get there with my full equipment. This, however, due to the heavy backpack takes 3-4 days instead of one. I can experience weather changes from quiet, warm and sunny days to noisy, windy, snowy, cloudy, stormy and freezing days and nights. Yes, even in summer! It may happen on Rainier, Shasta, Baker and any other mountain. One has to be prepared for such conditions, not just count on luck.
Once a year I take people (whoever wants to go) on the classic route of Mt. Shasta. I teach basic techniques such as how to climb w/o crampons different slopes, how to use ice-ax, and how to self-arrest (and anything you would like to know). Check my website, <a href="http://www.worldofadventures.com/marcus/" target="mrd"><font color="blue">worldofadventures.com/marcus</font></a> , for some descriptions and pictures and email me if you are interested. I'm planning next trip on 29/30 of May. Of course, it is free.
I climb (mostly solo): Rainier every year, Shasta a few times a year, and Whitney every year in winter.
By the way, in order to get a climbing permit on Rainier it has to be at least 2-pearsons team (no problem). For a solo permit, one has to be experienced (It's a totally different process.). If you are not roped, the ranger will ask you to rope to each other or to return. If you solo without a solo-permit, the ranger will ask you to return. That's my experience.