Are there two theories that are mathematically identical but ontologically different?

Copenhagen quantum mechanics and DeBroglie-Bohm quantum mechanics, mathematically they are equivalent, "metaphysicaly" or "epistemologicaly" they are quite different

update The reason they are equivalent is because they reach the same central equation (Schrödinger equation) but from different paths. So the rest computations and experimental results can be calculated the same and so on.

Special relativity and Lorentz ether theory (LET). From the linked Wikipedia article:

Because the same mathematical formalism occurs in both, it is not possible to distinguish between LET and SR by experiment. However, in LET the existence of an undetectable aether is assumed and the validity of the relativity principle seems to be only coincidental, which is one reason why SR is commonly preferred over LET.

The other answers bring up some nice examples of theories in physics that are equivalent in prediction but different in interpretation. I just want to bring up a trivialized example to illustrate the flaw in the reasoning in the question.

Suppose I have two theories which each predict the same quantity, say the maximum temperature at a given location on Earth on a given day. To make this concrete, suppose the first uses sophisticated modelling of the atmosphere, weather processes, etc. Suppose the second one is a sophisticated statistical algorithm which makes its prediction based on a large sample of historical data, comparing temperature trends across decades and a wealth of other data.

And now suppose that both theories produce correct predictions. At first this seems implausible given the complexity of the system and the radically different approaches taken, but it's not so far-fetched that both produce correct predictions within the errors on those predictions.

How can you tell which theory is "correct" (probably more fair to ask which theory is "more correct than the other")? You could try to extend both theories to be more general and predict more quantities, and see which one starts to break down (or more likely, both break).

I think the more interesting underlying question is "Is there a unique theory that accurately predicts all physical processes?". First, though, one needs to settle the question of "Can a theory of everything exist?", see Gödel's incompleteness theorems (and also this) for some information on that topic.