Value of the structure sheaf on a general open in Spec(R)

An even easier example than the two current answers is the following. (See also this MO answer.)

Example. Let $E$ be an elliptic curve over an algebraically closed field $k$. Let $O \in E(k)$ be the origin, and $P \in E(k)$ a non-torsion point. Let $X = E \setminus \{O\}$ and $U = X \setminus \{P\}$.

If $f \in S$ (i.e. $f \in R = \Gamma(X,\mathcal O_X)$ becomes invertible on $U$), then $\operatorname{div}(f) \subseteq E$ has to be supported on $\{O,P\}$. Because $P$ is non-torsion, the only way this can happen is if $\operatorname{div}(f) = 0$, i.e. $f \in k^\times$. We conclude that $S = R^\times = k^\times$, so $R[1/S] \cong R$. But $\Gamma(U,\mathcal O_U)$ is not isomorphic to $R$. $\square$

This is the canonical "affine open that is not standard affine open".

Here is the simplest example that I know. Let $R$ be $k[x,y,z]/\langle xy \rangle$. Let $I$ be $\langle \overline{x}(1+\overline{z}), \overline{y}(1-\overline{z}),\overline{z}^2-1 \rangle$. Then $S$ equals $k^\times$. Yet $A$ is the following countably generated $R$-algebra, $$A=R[u_n,v_n : n\in \mathbb{Z}_{\geq 0}]/J, $$ $$ J := \langle \overline{y}-u_0, (\overline{z}-1)u_{n+1}-u_n, \overline{x}-v_0, (\overline{z}+1)v_{n+1}-v_n, u_nv_n : n\geq 0 \rangle.$$

The map is not always an isomorphism (assuming there is no stupid mistake in my judgment). If you take an arbitrary commutative unital Noetherian ring and localize it at an arbitrary multiplicative set, the resulting ring is Noetherian (see here). If the map in question is always an isomorphism, then for a Noetherian $R$ and any open set $U\subset X=\mathrm{Spec}\,R$, the ring $\mathcal{O}_X(U)$ would be Noetherian. This need not be true. See Remark 3.7.b in this paper. You also may find this paper useful.