The big advantages are (a) they’re much more isotropic, (b) there is less effect from grid artifacts (reduced moiré patterns, etc.), (c) they are noticeably more efficient in terms of sampling rate for a given quality of signal. If everything is done on such a grid end-to-end, you get quite noticeably better quality results for just about everything (with the exception of slightly worse results for drawing sharp grid-aligned rectangles). Even swapping in such a grid in one or another part of the pipeline can usually net some advantage.

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.

Here's another great article about hexagonal grids:

http://www.redblobgames.com/grids/hexagons/

It's written with game design in mind, but many of the techniques are useful just the same.