What is the purpose of introducing the notion of points of value? The following is a quotation from Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(6.7 The Affine n-Space) .

For any $R$-algebra $R'$ let $\mathbb{A}^n_S(R')$ be the set of all $S$-morphisms $\mathop{Spec} R' \to \mathbb{A}^n_S$; this is the set of so-called $R'$-valued points of $\mathbb{A}^n_R$. Since the set of $R$-homomorphisms $R[t_1, \cdots , t_n] \to R'$ corresponds bijectively to the set $(R')^n$ of all $n$-tuples with entries in $R'$, via the map $\varphi \mapsto (\varphi(t_1), \cdots , \varphi(t_n))$, we see that the set of $R'$-valued points of $\mathbb{A}^n_R$ is given by
  $$ \mathbb{A}^n_R (R')=(R')^n.
$$

My question is ;
In this case, $(R')^n$ and $\mathbb{A}^n_R(R')$ coincide each other. Why the author introduce the notion of $R'$-valued points instead of $(R')^n$? When is the notion of point with value useful?
Thanks.
 A: Of course I can't read Bosch's mind but his notation might be preparation for a more general situation, namely:
Let  $I\subset R[T_1,\cdots,T_n]$  be an ideal  and define for any $R$-algebra $R'$ the subset $V_I(R')\subset \mathbb A^n_{R}(R')=(R')^n$ by the requirement $$V_I(R')=\{(r'_1,\cdots,r'_n)\in (R')^n\vert (\forall f\in I) \; f(r'_1,\cdots,r'_n)=0 \}\subset \mathbb A^n_R(R')       $$ 
Bosch has thus introduced a functor $$ \mathbb A^n_R :\operatorname {\mathcal {Alg}}_R\to \operatorname {\mathcal {Sets}}:R'\mapsto (R')^n$$ and the above is a subfunctor $$    V_I :\operatorname {\mathcal {Alg}}_R\to \operatorname {\mathcal {Sets}}:R'\mapsto V_I(R')$$    for which  we write   $V_I  \subset \mathbb A^n_R$.
A fundamental    point is that $V_I$ is representable by the $R-$algebra $A:=R[T_1,\cdots,T_n]/I$, which means that we have functorial bijections  $$Hom_{R-Alg}(A,R')\to V_I(R'):\varphi\mapsto (\overline {T_1},\cdots, \overline  {T_n}) $$ This opens the way to an even more general notion of points $ X(T)=Hom_{Schemes}(T,X)$  of an arbitrary, non-affine, scheme X with values in an other arbitrary scheme $T$ .
All this is  admirably explained in much detail in these notes (in French) by Ducros  .
A: From Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(7.2 Fiber Products)

First note that the universal property of fiber products just says that these products respect the formation of $T$-valued points, namely, 
  $$(X \times_S Y)(T) = X(T) \times Y(T)
$$
  for $S$-schemes $T$, recall that $X(T) = \mathrm{Hom}_S(T,X)$ is called the set of $T$-valued points of an $S$-scheme $X$. However, such a nice behavior cannot be expected from its underlying topological space. 

