Is there any non-trivial statement in PA + Con(PA), that was not already in PA? where PA = Peano Arithmetic
The standard model of PA (the basis of number theory) seems to be a model of PA + Con(PA) too. In this sense, is there a theorem in PA + Con(PA) that can't be proven in standard number theory?
This is not quite an answer but it is related. Gentzen showed that you can prove the consistency of PA using primitive recursive arithmetic together with transfinite induction up to a countable ordinal called $\epsilon_0$. As I understand it, he argued that the existence of $\varepsilon_0$ was less controversial than the consistency of PA so this was a good reason to believe that PA was consistent. So instead of considering adding $\text{Con}(PA)$ let's consider adding the existence of $\varepsilon_0$.
Then there are some "natural" statements such as Goodstein's theorem, the strengthened finite Ramsey theorem, or the winnability of the (Kirby-Paris) hydra game which can be proven using transfinite induction up to $\varepsilon_0$ (and which I believe are essentially equivalent to the existence of $\varepsilon_0$ , although I can't find a reference explicitly stating this) but which are known to be independent of PA. (If they're equivalent to the existence of $\varepsilon_0$ then this follows since PA can't prove the existence of $\varepsilon_0$, by Gentzen.)