This question and answer apply to Minecraft 1.4.7 and earlier. Minecraft 1.5 and later support high-resolution textures automatically, and do not exhibit the misplaced animations described below.
Minecraft versions before 1.5 were not written to support different-resolution textures.
The default Minecraft terrain texture in these versions is a 256×256 image with 16×16 tiles in it (a texture atlas).
The code which animates the animated textures in Minecraft (fire, water, lava, and portals) writes onto specific pixel offsets according to 16×16 tiles, which are wrong for larger textures, so you get small fire/water/portal images occupying other blocks. Presumably the patchers replace the 16× factors with 32×, or whatever fits the actual texture resolution.
The reason the textures mostly work is that the OpenGL 3D API used by Minecraft mostly takes coordinates in textures in terms of numbers between 0 and 1, rather than 0 and whatever the pixel size of the image is, so the main rendering doesn't care; only the animated textures, which are actually defined in terms of recalculating individual pixels and modifying the data of the texture in memory, are affected.
Also, tiles over a certain size (128×128, I think) will crash Minecraft. Below that size, you can still use HD textures and it will work except for the misplaced animations. The animated items will not animate, and (for a 32× pack) these will have animations on them: bricks, gold blocks, Netherrack, and pumpkins.
Neither and both. Confusing?
Here's the deal: Minecraft uses quite a deal of processing power, both with the CPU and the GPU. Texture packs are handed by the GPU.
Now consider the following: Loading an 8x8 texture is 1/4 of the load of loading a 16x16 texture (that is the default size). However (and here's the catch), despite the less work, you would only notice any difference if the GPU was bottlenecking the process. In other words, if you weren't running at top graphical performance, you may see some small difference.
The reasoning here is also valid for HD texture packs: the bigger the texture, the more load the GPU is subject to and your performance will drop accordingly.
Best Answer
Reversed textures is a bug in 1.7.2 that affects multiple North and East side textures. It's just way more obvious on beds than on most other blocks.
It's already marked as fixed in the bugtracker (so, fixed in Mojang's in-development code), so the fix will probably appear in the next update.