Why use colormap viridis over jet? As announced in https://www.youtube.com/watch?v=xAoljeRJ3lU, Matplotlib changes the default colormap from jet to viridis. 
However, I don't understand it pretty well. Maybe because I'm color blind?
The original colormap jet looks very strong, I can feel the contrast:

While the new colormap viridis lacks that contrast:

Can anyone please explain it simpler for me? I need the plot for my paper. And I need a good reason to convince my supervisor (and myself) that the viridis is a better one.
 A: The issue with using any kind of color scale to visually represent ordinal data is that of luminance monotonicity:  that is to say, if you have data that satisfy some kind of ordering relationship, that relationship should be reflected not just by changes in hue, but by luminance.  The problem with the "jet" color mapping is that the highest point in the mapping (corresponding to larger values) is given a red color, the middle range is given a yellow-green color, and the lowest is blue--but if we look at the perceived "brightness" (i.e., luminance) of those colors, it is clear that this mapping is not monotone.  The other color mapping in your question fixes this defect.
The reason for this property should be obvious, not the least of which is the fact that if such figures are reproduced in grayscale, interpretability is not lost.
A: See this video. You could also google it because there are a lot of (reasonable) jet-bashing everywhere.
Jet is very pleasing because it is flashy, colorful, and it does not require you to think about your color scale: even if you have just a few outliers, you still get "all the features" in your plot. You said it yourself: jet almost never lacks contrast.
However this comes at a very high price: jet literally shows things that do not exist. It creates contrast out of nowhere: just change your color scale a little bit in jet and you should see that the picture is change dramatically. Do the same thing in viridis, and you would merely have the impression that you are putting more or less light on the exact same thing.
If you don't like viridis, use the other colormaps that were discussed in the video above: they have the same nice properties, and they won't make your data lie. Also change the color scale: starting at 0, even if it is logical from a scientific point of view, may not be a good idea to represent these specific data (but change your colorbar to reflect that, e.g. "<25").
But again, see the video, there are a lot of examples in there as well as complete explanations.
A: You need the plot because you need to show data and you need a colormap because you know that the color you show will not be seen equally by all persons: any color is an interpretation through our visual perception.
Indeed, colors are subjective in the sense that they are interpreted by the brain (in the sense that a spectrum is transformed into a neural activity) into different levels of valence (or value) as a function of the colorbar given next to it. Your eyes will make a constant set of saccades to match the plot with the bar. 
JET is to be banned because it is perceptually ambiguous. One first feature of colors in visual perception is its value, that is the total brightness, that acts as the most direct feature. However, this value is non-monotonic in JET, such that ONE value in brightness may induce different values in perception. This is in particular true for the blueiash - yellowish tone (and that most of the time those that correspond to zero values) which artificially "pop up" from an image. Check such curves on :

Viridis (among other alternatives) is made to avoid that problem. You may read this full description for this choice, and how to adapt your colormap to the category of data to plot.
This should convince your supervisor.
A: There's several nice answers here already, but I think it's still pertinent to add another viewpoint, from the excellent paper

Good Colour Maps: How to Design Them. Peter Kovesi. arXiv:1509.03700 (2015). Software available here.

which lays out in a very clear fashion the principles of colour-map design, and provides a really nice tool to analyze them for perceptual uniformity:



This 'washboard' plot has a steady ramp from zero to one going left to right along the bottom, and the top of the plot has a sinusoidal modulation of uniform amplitude. For a properly-designed color map, all of the fringes at the top should show identical, or at least similar, contrast. However, when you put jet to the test, it is immediately obvious that this is not the case:



In other words, there are a ton of fringes, in the red and particularly the green stretches of jet, that get completely nuked out and become completely invisible, because the colour map simply does not have any contrast there. When you apply this to your data, the contrast in those regions will go the same way as the fringes. Similarly, the sharp contrasts along the bottom, on what should be a smooth linear scale, represent places where the map is introducing features that are not really present in the data.
