I have intuitively hard time understanding the result in mathematical logic that:
$\Phi^{\vdash}_{PA} \subset Th(R)$
$R$ here is $\{\mathbb N,+,*,0,1\}$, or in other words the standard model of natural numbers. The RH symbol means the theory of $R$ that is, the set of all provable sentences from $R$. The LH symbol is the set of all provable sentences from Peano Axioms. And Peano Axioms are formulated in terms of: $\{+,*,0,1\}$.
It's easy to see that all axioms of PA are derivable from $R$, so clearly $\Phi^{\vdash}_{PA} \subseteq Th(R) $. But once these formulas are derived from $R$, what even is the difference between the two at that point? They look exactly the same to me.
With the Beta-function lemma it can be proved that solving $Th(R)$ is equivalent to solving halting problem, so the set is not decidable and numerable, while the other set is. But it's still difficult to see what's the difference between the two. Perhaps someone can provide an intuitive example?
(I believe an equivalent proof would be that $Th(R)$ is complete, while $\Phi^{\vdash}_{PA}$ is not).
I believe there's a misunderstanding here. "$\Phi_{PA}^\vdash$," I believe, is the set of sentences provable from PA, while $Th(R)$ is the set of sentences true in the structure $R$, contrary two what you've written (incidentally, using "$R$" to denote the structure $(\mathbb{N}; +, *, 0, 1)$ is terrible). It looks like you've swapped the RHS and LHS in your explanation of what they mean.
So the statement you're trying to prove is:
Every sentence that PA proves, is true in $R$.
But there is a sentence true in $R$ which PA doesn't prove.
As you've observed the first is immediate, so all the weight is on the second.
At this point it's useful to point out the difference between provability from a theory and truth in a structure (in particular, phrases like "provable sentences from $R$," "formulas are derived from $R$" treat $R$ like a theory, which it's not). If $\varphi$ is provable from a theory $T$, then there is a witness to this, namely a proof. In particular, the set of sentences provable from any (recursive) theory $T$ is recursively enumerable.
By contrast, there is no reason for the set of sentences true in a structure to be recursively enumerable! Roughly, the "reason" that a sentence is true in a structure may not be "finitely describable" in any way. (And when you write "With the Beta-function lemma it can be proved that solving $Th(R)$ is equivalent to solving halting problem," this is incorrect - while it is true that the halting problem can determine whether a recursive theory proves a sentence, since $Th(R)$ isn't recursively axiomatizable that doesn't help us at all.)
In fact, this is the case here: the set $Th(R)$ of all sentences true of $R$ is not recursively enumerable, hence cannot be equal to $\Phi_{PA}^{\vdash}$, which is recursively enumerable; and so this, together with the first bullet point, proves the second.
So now you just need to show that $Th(R)$ is not recursively enumerable. This is a good exercise - try to, for instance, explicitly code the set $0''$ into $Th(R)$. You'll show in this way that $Th(R)$ computes $0''$, and hence cannot be recursively enumerable.