Real space term for PPPM vs PME
There are a number of ways of looking at Ewald's summation, but one is that instead of trying to solve the original problem where the potential is due to a set of point charges you, instead, solve the system for a set of smeared charges and add a correction. Now there are limitations on what are good ways to do the smearing, but there are still quite a lot of possible choices. The most common is to replace the original delta function charge with a Gaussian, the width of which is determined by alpha in your question. This leads to the error function and exponentials you are used to. However it's not the only possible choice. In equation 22 in the paper you refer to a different way of smearing the charges has been chosen, a linear ramp centred on the particle. This is described in equation 21. As the smearing is different the individual terms in the Ewald summation have different forms when compared to Gaussian smearing, but it's no less valid a way of proceeding.

If you want to learn more about this look into Fourier Transforms and the Convolution theorem. A bit of reading on that and the Ewald Summation in all its forms becomes pretty straightforward.