Use marginal_likelihood when your observed data is drawn directly from the GP. If you are using the GP as a prior distribution for latent (i.e. unobserved) variables, then use prior. Here’s an example from PyMC v3 which shows how the prior method can be used to create latent RVs. In that example, the observed data are discrete counts (which clearly cannot be drawn from a standard GP) but we use the GP to place a prior over the log-mean value of the Poisson distributed counts.
https://docs.pymc.io/en/v3/pymc-examples/examples/case_studies/log-gaussian-cox-process.html