Why are there sensors that encode their readings as serial sequence of PWMs?
the more conventional 4 to 20 mA 


More conventional is a very relative term, and it seems you might be coming from a process control background, where sensor signals are often processed and converted to a current internally in the sensor. Let me assure you that it's not the most common thing in the world.


  What's the advantage […] rather than […] converted to digital at the receiving end?


Well, having an ADC in the microcontroller is a luxury. The sensor in question has a 14 bit output; finding a microcontroller with a 14 bit ADC will increase your material cost.


  Is this more accurate?


Yes. 14 bit means \$2^{14}\$ possible values. Let's say the voltage signal would have a full amplitude range of 0 V to 5 V. In that case, your voltage step would be \$5\cdot2^{-14}\,\text V\approx0.3\,\text{mV}\$. That's very little! Interference, temperature variations and noise in your ADC will be a multiple of that, unless you can very closely control a lot of things, which will make your system very, very complex and expensive.

So: Whenever you need digital values at the end, convert analog to digital as early as technically feasible.


  Alternatively why not go HART / PROFIBUS etc. ?


Because buses like these are really complicated to implement on both the sensor and the controller, and if you're really just attaching the sensor to a microcontroller on the same PCB, why make things complicated, and costly?

The four-states state machine I showed that is able to receive this kind of signal might seem complicated to someone not overly used to embedded development, but imagine you'd have to write a full PROFIBUS endpoint. Good luck!