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Depending on what you're producing, working in 8bit will probably be fine. 32BPC (Floating Point) is primarily for High Dynamic Range footage and effects.
For a run down on some math.. 8BPC means you have 256 shades of each color. That number is computed by 2^8 (2 to the 8th power, 2*2*2*2*2*2*2*2). Then if you take 256^3 (256R*256G*256B), that's where "Millions of colors" is derived from.
16 and 32BPC work the same way (2^16, actually, I'm not sure about 32, cause that's floating point, ie you can have decimal places, such as 78.56). Now, if you think of 8BPC as normalized (0-1), 16 and 32BPC work by allowing values greater than 1. So you can generate overblown highlights and such.
A lot of times, if you see banding in your images (such as with Generate/Ramp), that can be alleviated by changing your project to 16bit).
Just enabling 32BPC will not make LDR (Low Dynamic Range) standard footage look better. Adding effects and blend modes is a good way of seeing the difference between 8, 16 and 32bpc.
>Does it result in a significantly better video and image?
Not necessarily. Pixel processing operations are different which is mostly noticeable in things like motion blur or layers and effects that simulate "light" (Glow, Add mode).
>I kinda assumed that projects in HD format would have simply been in
>this higher quality mode already.
As said above and explained by David - it has no relevance in terms of final output quality as such. It may prevent banding, but 16bpc can do that as well. 32bpc has other benefits like retaining "super-whites" and the already mentioned blending.
>Does it really slow render times?
It does due to the deeper processing.
>16 and 32BPC work the same way
Nope, absolutely not. Float processing is completely different. It's more about matching binary patterns than actual logical math in the classical sense. In theory the numeric resolution is infinite, but most apps will stop at 2^96 or 2^128.
For a description of the tradeoffs of working in higher color bit depths, see the "Set the color depth" and "High dynamic range color" sections of After Effects Help on the Web. The latter includes a link to a post by Jonas that goes into quite a lot of detail about working in 32-bpc color.
> Should also play with the "working space" option there as well?
That control is for color management, which is a different beast. For information on that control, see the "Choose a working color space and enable color management" section of After Effects Help on the Web. You should probably start with "Why you should use color management" before you go fiddling with the color management controls, though.
Check out this fun video on how dramatically better 32-bit color is when used with glow and Trapcode Shine: