This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field.

When one has a polynomial subset $V$ of $R^n$, the following two are equally sensible ways of putting a structure sheaf on $V$:

One is by considering regular functions in the sense of usual scheme theory: in this case the global regular functions are $R$-polynomials in $n$ variables modulo the ideal of those polynomials vanishing on $V$. If, more precisely, we call $X$ the corresponding $R$-scheme (with all of its non-$R$-points too, which by the way are reconstructible from the set $V\subseteq R^n$) and $O_X$ its structure sheaf, then $X(R)=V\subseteq R^n$ and $O_X(X)\simeq R[x_1,\ldots, x_n]/I_X$.

The other way is by declaring that a regular function is a ratio of polynomials with nonvanishing denominator. We will call such functions $R$-regular, and $R_V$ the resulting structure sheaf. We call $(V,R_V)$ an $R$-algebraic variety. This definition seems to be standard in real algebraic geometry, see e.g. Bochnak-Coste-Roy - Real algebraic geometry (Section 3.2). I think it doesn't change much if we consider the topological space $X$ of the scheme in point 1) instead, endowed with the sheaf $R_X$ that sends an open set $U$ to the rational functions on $U\subseteq X$ that are regular at each point of $U\cap X(R)$.

The resulting structure sheaves are not the same. For example, consider the real line: the function $\frac{1}{1+x^2}$ is an $R$-regular function which is not (scheme-theoretically) regular.

Likewise, one can define abstract $R$-algebraic varieties, and $R$-regular maps thereof.

The curious thing is that every projective $R$-algebraic variety is $R$-biregularly isomorphic to an affine one. Indeed, the set theoretic map (example 1.5 in Ottaviani - Real algebraic geometry. A few basics or theorem 3.4.4 in BCR) $$\mathbb{P}^n(R)\to \operatorname{Sym}^2(R^{n+1})\;\;,\quad (x_0:\ldots : x_n)\mapsto \frac{x_ix_j}{\sum_{h=1}^n x_h^2}$$ is an $R$-regular embedding. This does not correspond to an everywhere-defined morphism of schemes, as is immediately seen by looking at any component of the map in a standard affine chart of $\mathbb{P}^n$.

Are there non-(quasi-)affine abstract $R$-algebraic varieties at all?

Edit: I think the "quasi" in "quasi-affine" may be pleonastic: I haven't checked the details but a quasi-affine $R$-algebraic variety should very often be affine. Indeed, if $X=W\smallsetminus Y$, $Y\subset W \subseteq R^n$ with $W$ affine and $Y$ closed (maybe with some assumptions on $Y$), the real blowup $\operatorname{Bl}_Y W$ is closed in some $\mathbb{P}^{m}\times W$ and the latter is *affine*; but now the "missing" set $E$ has become a divisor: $X\simeq (\operatorname{Bl}_Y W)\smallsetminus E$, and affine minus a divisor is still affine.

The above example (the one of projective space embedding in an affine space) shows that the category $\text{$R$-Var}$ of $R$-algebraic varieties is *not* a full subcategory of schemes over $\operatorname{Spec}(R)$. On the other hand, I think the category $\operatorname{Sch}'_R$ of finite type separated reduced schemes over $\operatorname{Spec}(R)$ is a full subcategory of $\text{$R$-Var}$. [Edit: following the comment of Julian Rosen, we probably also want to require the schemes in $\operatorname{Sch}'_R$ to have dense $R$-points]

Are there two non-isomorphic schemes in $\operatorname{Sch}'_R$ that become isomorphic in $\text{$R$-Var}$?

Edit: even before posting, I found example 3.2.8 in BCR. There is also proposition 3.5.2 in BCR, the $R$-biregular isomorphism between the circle $x^2+y^2=1$ and $\mathbb{P}^1_R$. And between the "quadric" sphere and the "Riemann" sphere (i.e. *complex* projective line thought of as a *real* algebraic variety).

In which other ways does $\text{$R$-Var}$ deviate from $\operatorname{Sch}'_R$?

Note: I'm *not* asking how real algebraic geometry deviates from complex algebraic geometry (which is surely addressed in a preexisting MO question).

Edit: (added following question)

For non real-closed fields, or fields of positive characteristic, do people consider varieties in the sense of 1) or in the sense of 2)?

For example, should $1/(1+x^2)$ be a regular function on the line over $\mathbb{F}_7$? (It's a well defined function on a finite field, so there will be a polynomial realizing its values set theoretically, but should it be enough?) -- Or, should 1/(x^2-3) be a regular function on the line over $\mathbb{Q}(\sqrt{2})$?