Lots, if you're into printing. Banding that's invisible on a monitor can become a monster when printed. An expensive monster mistake if you're printing a poster sized giclee.
This is a big reason why Photoshop dithers by default when going from a higher range down to 8 bits per channel.
Another reason the need for dither might be even increasing today is that digital cameras have only very recently gotten better than an 8 bit signal to noise ratio. We've had 16 bit RAW formats for a long time, but most cameras still had noise that was greater than 1/256. I tested one of my old Canon EOS cameras and found even in very well lit conditions, it's signal was a little over 5 bits, even though I could get a 16 bit image file out.
With the newest cameras, the signal to noise is sometimes exceeding 12 stops, so photographers might be running into banding issues in RAW images more often. Except a lot of them never notice because Photoshop quietly handles it.
A lot of artifacts are easier to deal with on a hexagon-pixel grid. Halftoning and error-diffusion dithering work really well.
I'd guess that the need for error diffusion and half-toning might wane in the future. (Not sure about that, but that's what I'd guess.) Are there other benefits you get with hex images?
The downside is that there are a zillion implementations of everything you can imagine w/r/t a square grid, whereas implementations on a triangular grid of hexagonal pixels are fewer and further between, often some crappy matlab code written by a grad student, so you’d maybe need to reimplement, optimize, debug, etc. a whole pile of tools, or even invent some new algorithms for doing stuff that hasn’t been tried on such a grid before.
There have been lots of papers written about the topic, and at least one book. http://www.springer.com/us/book/9781852339142
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It’s similar to the advantages you get from tri-axial fabric: it’s better but requires custom machinery, so in practice is only used for bullet-proof stuff, space suits, and art projects. Hexagonal pixels end up getting used in medical imaging, astronomical imaging, and the like, where pixels are individually expensive.
http://www.redblobgames.com/grids/hexagons/
It's written with game design in mind, but many of the techniques are useful just the same.
I typically use Sierra Lite.
I've recently been experimenting with error diffusion along continuous space-filling curves (I forget where I saw this technique first), and using color difference functions (CIEDE2000) to gauge quantization error. These are overkill, but it sure beats doing it the same boring way forever.
Frankly, I would like it if the CRTCs attached to modern GPUs had some ability to apply dither (even just ordered dither) to buffers as they're scanned out, while exposing more precision to applications.
I found the thing which inspired my space filling curve exploration: https://www.compuphase.com/riemer.htm
The Butteraugli score between the standard lena and the Riemersma dither is about 0.5 better than the standard Floyd-Steinberg one.
Approaches like this come closer to back-propagation of errors, the lack of which is where some of the artifacts of unidirectional dither come from. In practice, any decent (full bleed) error diffusion dither works perfectly fine going between 16 and 8 bit planes, but why shouldn't I try some things out which (as you are quick to point out) may have very little value.
Heck, even gamma correction isn't strictly necessary when you're going from 16 to 8 bit sRGB.
Why do you think this Riemersma method does better at that metric? Maybe it just keeps the diffused errors spatially closer to the original pixel?
I can understand that it would become less useful, because error diffusion does less if you have more levels. But (I expect) it's still useful at 6 bits (64 values), is 8 not just an arbitrary treshold?
8 bits is not just an arbitrary threshold, it's the number of bits you need so that a 1 bit difference is just about the "just noticeable difference", assuming you've handled gamma and color reasonably well. 9 bits is more lightness resolution than you can normally see, and 7 bits is too few. If you want to geek out super hard, check out figure 238 here: http://www.telescope-optics.net/eye_intensity_response.htm
The JND of Weber's law is 0.4 - 0.5% for print and monitors most of the time, which is between 1/200 and 1/250. This is why 8 bits is so super popular, it has a 1-bit step of 1/256, which is normally just barely below what we can see with our eyes.
You may need to dither at 6 bits, banding could start to become pretty obvious. Whether you need error diffusion, I don't know, I doubt it would always useful at 6 bits, but I'm sure it would sometimes. I'd say try it on some images and compare to random dithering. Random dithering only sucks on really small palettes like 2 or 16 total colors. It's not that bad on a 256 color gif, and it becomes hard to tell the difference when you're above 4 bits per channel or 12 bits per pixel. 6 bits per channel is decent color resolution, and random will sometimes be good enough. If it's a photograph from a digital camera that's older than a year or two, chances are pretty good you can threshold the image without even dithering, because it's already noisy enough.
A few years ago there was a need for 15-bpp or 16-bpp color images on phones rather than the 24-bpp images we usually work with, and dithering was a great way of producing them. No idea how much need there is today though.