My question is rather simple for an algebraic-geometer maybe and because I'm not, it confuses me a lot and has to do with the following. Sometimes, I see authors indentify an affine variety $V \subset \mathbb{A}^{n}$ over an algebraically closed field say $\mathbb{K}$, with the spectrum $Spec(\mathbb{K}[V])$, and they are referring to the latter as the the affine variety with coordinate ring $\mathbb{K}[V]$. (where as far as I know the last spectrum it is called affine scheme in the literature)

So, it's obvious by definition that those two sets are not equal, hence it's about an identification (natural or if you prefer categorical) between them. Even if $V$ is irreducible variety (which means that we don't have other primes except the maximals and the trivial $\{ 0 \}$ and due to Nullstellenstaz we have a "good" correspondence between the points of $V$ and elements in $Spec(V)$) there is something missing, namely the generic point we add because of the non-maximal prime ideal $\{ 0 \}$. I would understand the identification of $V$ with the maximal spectrum $mSpec(\mathbb{K}[V])$ in that case but the previous one doesn't make any sense. Can you please explain me how this identification comes in and what's the idea behind the "equality" $V=Spec(\mathbb{K}[V])$?