How does hPES compare to the learning rates of ANNs?
According to the paper, the advantage of this new approach over conventional ANNs, Deep Belief  Networks (DBN) and Self-Organising Networks (SON) are:


  
  Remains functional during online learning.  
  Requires only two layers
  connected with simultaneous supervised and unsupervised learning
  Employs spiking neuron models to reproduce central features of
  biological learning, such as spike-timing dependent plasticity
  (STDP)
  


So arguably, hPES is superior to ANNs in terms of capability, but in terms of performance, you'll have to compare the code from his experiments with ANNs meant to accomplish the same task, but since both methodologies are designed with different purposes in mind, it might not be worth comparing. Additionally, it should be noted that although the author claims to have created a SON, the validation of this claim in the paper (and his master's thesis) is pretty weak. In the paper he only proves that the SON increases sparsity, which is not what SONs (for example Kohonen networks) are generally used for.

Finally, note that although hPES is more biologically plausible, it still has some of the same problems that ANNs have. Namely, the parameters that it takes in must be optimized for the specific task it's learning (some approaches for solving this with ANNs include Genetic Algorithms) even if it is less sensitive to parameter modification. The author mentions this investigation of parameters as being part of future work.