Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter).

I have questions regarding the difference between the two:

I have some intuition about primitive recursive functions. For example, a function is primitive recursive if its algorithm is described by means of "only for-loops, not while-loops". How the intuition for $\Delta^0_0$ relations are different from that for primitive recursive ones?

What syntactic condition does primitive recursiveness correspond to, if it does at all? More precisely, if $R$ is a primitive recursive relation, what is the syntactic necessary and sufficient condition for $\phi$ if $\bar n \in R \Leftrightarrow \mathbb N \models \phi(\bar n)$, modulo first-order equivalence of $\phi$?

EDIT: The for-loop explanation of primitive recursion can, for example, be seen in Section 2.5 of Schwichtenberg and Wainer's *Proofs and Computations*.