I would say that HBM is certainly "more Bayesian" than EB, as marginalizing is a more Bayesian approach than optimizing. Essentially it seems to me that EB ignores the uncertainty in the hyper-parameters, whereas HBM attempts to include it in the analysis. I suspect HMB is a good idea where there is little data and hence significant uncertainty in the hyper-parameters, which must be accounted for. On the other hand for large datasets EB becomes more attractive as it is generally less computationally expensive and the the volume of data often means the results are much less sensitive to the hyper-parameter settings.
I have worked on Gaussian process classifiers and quite often optimizing the hyper-parameters to maximize the marginal likelihood results in over-fitting the ML and hence significant degradation in generalization performance. I suspect in those cases, a full HBM treatment would be more reliable, but also much more expensive.