The second order arithmetic of the naturals is stronger than PA because it allows induction on arbitrary subsets of the naturals, while PA only allows it on subsets that are an extension of some predicate, the latter collection is countable and the former is uncountable.
But any subset I can think of is in the latter, e.g. all r.e. sets only involve one first order quantifier, and you can have as many quantifiers as you like. Can one construct an explicit example of a subset of the naturals which is not the extension of a predicate in PA? If so please provide one.
Note: I think the answer is no, because I remember reading that Markov school of constructive maths adopted the 'false church's thesis' that ${Rec}=\mathbb{N}^\mathbb{N}$, if this is the case, I would be interested in any references to a proof of this. If this isn't the case, perhaps someone more learned could explain what Markov was up to...
Well, letting $\#$ be the usual Godel numbering function, the coded true first-order theory of arithmetic $\{\#(\varphi):\varphi\in\mathsf{FOL},\mathbb{N}\models\varphi\}$ is second-order definable but (per Tarski) not first-order definable.
For a more "structural" example, you can look at Kleene's $\mathcal{O}$, which is essentially the set of codes for computable well-orderings. Again, this is not first-order definable in $\mathbb{N}$ - in fact, it's much more complicated than the previous example in a precise sense, even though at first it may seem more concrete!