Do EEVEE samples really matter when rendering?
Can't really see a difference except for render time
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There are several things affected by EEVEE sampling, as you can see in this example scene, rendered at 1, 8 and 64 samples:
soft shadows are obtained by sampling large light sources using many small light sources. Low samples will "reveal" the point lights. This is generally the main downside of a low-sample rendering, therefore you should choose the number of samples based on how good your soft shadows look.
the direction of refracted and reflected is also sampled because, especially around the corners, a sub-pixel change in the direction of the ray can make a difference between hitting an object, another object, or the sky. For low samples, this results in a strange pixelated blur
aliasing is what happens when a curved profile is drawn using pixels: it gets jagged if the pixels aren't "smoothed" (antialiased) adequately. More samples make the contours smoother.
[Added Sept 2020] the quality of Alpha Hashed transparency is also affected by the global sampling. Indeed, for each sample, a pixel will either be rendered as fully opaque or fully transparent: a fractional transparency is a result of averaging over multiple samples. With too few samples, tranparency appears patched/noisy:
[Added Sept 2020, suggested by Andrey Sokolov] the number of volumetric slices rendered depends on the EEVEE sampling: note how at 1 sample the dust cloud appears 2-dimensional.
PS There is another "Samples" count inside the Volumetric tab that controls the quality of each slice, and is independent of the global EEVEE sampling.