"The notion of an affine algebraic set is still not satisfactory" In their book "Algebraic Geometry I" Gortz and Wedhorn at page 16, after defining the category of affine algebraic sets (Zariski topology in $\mathbb A^n_k$, corrispondence between radical ideals and closed subsets, morphisms of affine algebraic sets ecc...), say:

The notion of an affine algebraic set is still not satisfactory. We
  list three problems: 
  
  
*
  
*Open subsets of affine algebraic sets do not
  carry the structure of an affine algebraic set in a natural way. In
  particular we cannot glue affine algebraic sets along open subsets
  (although this is a “natural operation” for geometric object
  
*Intersections of affine algebraic sets in $\mathbb A^n_k$ are closed and hence
  again affine algebraic sets. But we cannot distinguish between $V (X) \cap
V (Y ) ⊂ \mathbb A^2_k$ and $V (Y ) \cap V (X^2 −Y ) \subset \mathbb A^2_k$ although the geometric
  situation seems to be different (we will see similar phenomena later
  when we study fibers of morphisms)
  
*Affine algebraic sets seem not
  to help in studying solutions of polynomial equations in more general
  rings than algebraically closed fields. 
The first problem is due to the fact that affine algebraic sets are necessarily      embedded in an affine space.

Now, for me the third problem is obviously clear. But what is the meaning of the first two problems? I need some further explanations.
 Moreover, why they say that the problem 1. is due to the embedding in $\mathbb A^n_k$?
 A: 1) Everything manifold can be embedded in one $\mathbb R^n$ ; this is false in algebraic geometry. For example, if I take $\rm N$ copies of $\mathbb R^2$ and that I tell you how I decide to glue them, I get a manifold (if I glue well). This manifold can be embedded in $\mathbb R^4$, so it is affine.
In Algebraic Geometry, if I give you affine schemes in $\mathbb A^2$ and that I tell you how I glue them nicely, then the result need not be embeddable !
2) For this problem, there is a very nice way to see it. Consider $\mathrm{V(X( X} - a))$ in $\mathbb A^1$. It has two points if $a \not = 0$. Then let $a \to 0$, you will get $\mathrm{V(X^2)}$ which has only one points, but which has to be thought as two points infinitessimaly close.
In other words, the theory of schemes will allow you to make a difference between $\rm V(X)$ which represent only one point and $\rm V(X^2)$ which represent the infinitesimal process of two points coliding and becomming infinitessimaly close.
A: (1) It can be shown for instance that $\mathbb A^2\setminus \{(0,0)\}$ is not an affine algebraic set in any affine space $\mathbb A^n$. 
(2) Both intersections are reduced to the origin $(0,0)$. But clearly, in the first case, the curves $V(X), V(Y)$ meets transversally, while in the second case, both curves share the same tangent line.
