If you want to represent a one-dimensional gradient where what is most important is "hot vs cold", i.e. the extremes, then you shouldn't be using a colormap (rainbow) in the first place, you should be using a direct gradient instead, whether grayscale or interpolated between two nearby colors (e.g. red/yellow).
Colormaps are for where highlighting local variation is important -- more important than indicating whether something is closer to one extreme or another. In the examples, this means being able to distinguish the patterns of the objects. In math plots, it means helping to see the shapes of curves. In weather maps, that means judging variation within your state, even while the map ranges from Alaska to Florida.
For this, it's most important to be able to maximize contrast in local differences but also have the perceptual rate of change be roughly comparable.
Rainbows maximize contrast using hue, because our eyes can simply distinguish so much more than with just brightness. Turbo makes the rate of change more constant than Jet. And Jet/Turbo most importantly double the resolution that Inferno has, going from dark-bright-dark rather than just dark-bright.
https://www.bioss.ac.uk/people/chris/colorpaper.pdf
EDIT the "hue ramps" style of colormap is also quite interesting, although i can't find an example, and can't be bothered to create one:
http://soliton.vm.bytemark.co.uk/pub/cpt-city/imagej/tn/hue_...
I find Inferno/Viridis perfectly good for representing one-dimensional gradients. They're easier to glance at then a single hue and show contrast between min/max better. Greyscale and single hue can also make it hard to notice small variations.
I can see Turbo being useful for heatmaps where you want to notice differences rather than variation between hot and cold.
In general though I would say that choice of colormap is subjective and there isn't a general rule for when to use which colormap. I usually plot the same image in multiple colormaps and choose the one which helps me the most for that usecase. Another person may choose differently.
That’s basically the objection this mapping is aiming to address.
Where jet and other non-uniform maps are useful is in highlighting small fluctuations in the value. It's less about z1 > z2 and more about z1 != z2 across the image plane.
A similar problem happens with datashader in histogram normalisation mode. It's great for highlighting small variations in the underlying density and revealing detail that you'd otherwise never see, but worthless for trying to compare one density to another by eye.
Horses for courses.
Computers are already really good at dealing with large-scale details. The human eye can notice edge-case scenarios in a large dataset pretty rapidly. I'm sold on Turbo because it lets your eyes change scale on the fly. You can be looking at the large scale and rapidly focus in on particulars within a smaller band, and back out again, as needed.
I've tried inferno and viridis to visualize weather radar data, and... rainbow-like colormap simply works better. And this is true for most weather data like wind or temperature. With a uniform colormap, you wouldn't be able to see local variations, e.g. differences in temperature within the UK.
On this point I don't see Turbo is being a do it all colormap. In the depth comparison images I still see Inferno and Viridis being superior and personally have no confusion between which spheres line up with which rings. Turbo in fact confuses me on this by introducing the false 2D color gradient.
Where I see Turbo being useful, as others have pointed out, is for heatmaps. I personally find Inferno's low end contrast being problematic and usually resort to Spectral for picking out details. If you can live with the fact that the lightest part of the chair in the test image is rendered darkly in Turbo (compare with Inferno), then Turbo is great for purely picking out detail (and taking brightness with a pinch of salt).
In the end, it depends on your usecase. Use a linear colormap when appropriate and vice versa.