Playing as a Bosmer, I got the sense that there wasn't really any point to using my racial power once I got the Animal Allegiance shout. Was I missing something about how much better this power is, aside from the obvious difference that you can switch from the power and use a shout immediately after? And along similar lines, is there any advantage to ever using the Nord's racial power once you get the Dismay shout?

No it's an joke, only time its useful is if you run into an strong animal like a beer at low level.
First something you can use once a day is pretty useless, you can not use it often so you use it so rarely you forget it.
Secondly it only work at very low level.
The lesser powers like the Khajiit night eye is nice, resistances is also nice,

Nords and Bretons have pretty good ones (both active and passive).

Water breathing is nice. One less spell/enchantment I need to learn.

Some are better than others that's for sure. The ones I like most are the +resist ones, especially the poison resist from Bosmer and Redguards are ace in my opinion.

Bretons and Orcs have great racial abilities. High elves are decent as long as you wouldn't normally exploit the magicka system via enchanting. The only other race with powers I found useful were Redguards when playing Morrowind and Oblivion, but not in Skyrim. They svck in Skyrim.

I recently read that the Nord's fear power has a magnitude of 99, which is very nice. Now that I think of it, it may be among the strongest illusion spells in the game if the Nord becomes a vampire.

To be clear, I actually found a lot of the other powers more useful; it was specifically the Bosmer and Nord powers that seemed to be rendered useless by shouts later in the game. Other daily powers require either items or magicka to accomplish (e.g., Argonians' health regeneration or Redguards' stamina regen), or aren't even duplicated by other triggerable abilities at all (like the Orcs' dual damage dealing/damage regen, or even the Imperials' area-effect, level-ignoring Calm effect). Just wanted to make sure I wasn't missing anything. I suppose the late-game uselessness of these powers are offset by the fact that frost resistance is HUGE, and Bosmer get both poison and disease resistances, whereas other races tend to get only one passive resistance.

The Nords are half the people they used to be in Morrowind. They were once Immune to Frost and partially resistant to Shock, and now they are only 50% resistant to frost.

Sissies.