Naive Bayes vs Full Bayes model classifiers
There's no clear definition of "Full Bayes" as a classifier. Most "real world" non-Naive Bayesian classifiers take into account some but not all dependencies between features. That is, they make independence assumptions based on the meaning of the features. 

If by "full Bayesian" you mean a joint model (as your example suggests), then one of the problems is that such a model doesn't generalize: it just describes the probabilities in the training set, and that implies that it's likely to overfit badly. This is actually why NB works quite well in most cases: yes it makes unrealistic independence assumptions, but this simplification allows the model to capture basic patterns from the data. In other words, the ability of the model to generalize comes from its excessively simplified assumptions.

Note: as far as I can tell, your example is well chosen and you should see a big difference between NB and a joint model: NB should perform no better than a random baseline while the joint model should obtain near perfect accuracy. There's probably a mistake somewhere if you don't obtain these results. But while this is a good toy example, it cannot help you understand the advantage of the NB assumptions.