Yes, comparing these models is totally straightforward :)
In fact, when we compare continuous models to discrete models, what we are doing is finding a sensible discretization of the continuous model in order that we can treat the comparison as a comparison between to discrete models. The important thing is to make sure that we are comparing probability mass to probability mass (discrete models, including discretized continuous models), or probability density to probability density (continuous models). What we cannot do is compare probability density to probability mass.
Aki’s notebook is not pointing out an exception to this; rather, it’s pointing out that in many cases the discretization of a continuous model to a discrete model is numerically trivial to do, so that we can form the log probability density matrix for the continuous model, and what we create will be numerically identical to a good approximation to the log probability mass matrix of a discretized version.