Is gamma still important to take into account? Do different monitors (including mobile screens) still use significantly different gamma functions when displaying colour images? Is there a standardised way to represent colour that can then be translated according to the gamma of the monitor, or are they all sufficiently similar to preclude the need for this nowadays?
Yes, while many screens and OS operations are using a gamma of 2.2 your hardware and computation result still need to be corrected. There are also special mediums such as broadcast TV's that have a different gamma. Sensor equipment like cameras are mostly linear so they need to be adjusted accordingly. Besides this the user can set their system to whatever gamma they like. 

In computer graphics primary reason to account for gamma is that your computation is most likely done in linear space* or summing light contributions becomes unnecessarily complicated. In any case gamma is really a simplification of things, you'd be much much better of doing profile to profile conversions if its possible to invest the computational time to do so.

Mathematical explanation

This may be a simpler to understand. Your monitor has a display error of $g(x)$ which is no problem if your data is conforming to $g(x)$, however if you want to calculate:

$$ a+b+c+d $$

and $a$, $b$, $c$, $d$, have the error of $g(x)$ you would actually need to calculate.

$$ g(g^{-1}(a)+g^{-1}(b)+g^{-1}(c)+g^{-1}(d)) $$

You would need to do this for each sub element over and over again. Therefore instead of running the transform each time you transform all once to linear and then once back. 

And finally a artistic reason

It may just look better with a different gamma. Many games have a gamma adjustment for this reason.

* Worst case is that you think your computations as linear but do not compensate for the monitors nonlinear output charachteristics.