We sailed on 3/25 on the Sapphire, which was a crowded spring break trip. One of the anytime dining rooms had to be designated traditional for an early seating, but then became open to the rest of us.
Oftentimes, we would choose a restaurant only to find that we could not be seated at that time. They would give us a beeper (like at Olive Garden) and we would then drop by the other restaurants. If they could seat us immediately, we would give the beeper to the maitre d' and he would cancel it out.
The menu changed each night, and matched the menu in the traditional dining room, with one exception. Each restaurant offered one entree that was specially created for that themed restaurant (i.e. Italian, asian, southwestern, etc). That may have been the same for the entire week. I can't say for sure.
One thing you will miss when dining in the anytime resaurants is the parade of the flaming baked alaskas. They only do that in the main dining room. Also, if you are accustomed to the singing entertainment by the waiters on RCL, you won't find that on Princess. FWIW, there were no midnight buffets or chocolate extravaganzas.