I recently learnt about correct and stable cardinals and ordinals.
A cardinal $\kappa$ is called $\Sigma_n$-correct iff $V_\kappa \prec_n V$.
An ordinal $\alpha$ is called $\Sigma_n$-stable iff $L_\alpha \prec_n L$.
Apart from one being a cardinal and the other being an ordinal, is there really any difference between the two?
To start with, note that stability is a very, well, stable notion: since the $L$-hierarchy is appropriately absolute, if $\alpha$ is $n$-stable in $V$ then $\alpha$ is also $n$-stable in every generic extension $V[G]$. This is very false for correctness (this is a good exercise - think about e.g. starting with $V=L$ and then forcing a failure of $\mathsf{GCH}$ above what $V$ thinks is the smallest $2$-correct cardinal).
So even at a very coarse level - "How much can these notions be affected by forcing?" - we see a fundamental difference. Relatedly, by Tarski's theorem the class of pairs $(\alpha,n)$ such that $\alpha$ is $n$-correct is never (parameter-freely-)definable in $V$, but the class of pairs $(\alpha,n)$ such that $\alpha$ is $n$-stable may be definable in $V$ (if $V$ is very far from $L$ - e.g. if $V\models$ "$0^\sharp$ exists").