$\DeclareMathOperator\Diff{Diff}$Suppose for simplicity that $X$ is affine, it is then possible to define $\Diff(X)$ — the ring of Grothendieck differential operators. When $X$ is smooth, then



> 
> **Definition.** the category of $D$-modules on $X$ is defined to be modules over $\Diff(X)$. (Category **1**)
> 
> 
> 


However, when $X$ is singular, this is not the right category to consider. One usually follows Kashiwara's approach:



> 
> **Definition.** choose a closed embedding $X\hookrightarrow V$ and define $D$-modules be to modules over $\Diff(V)$ such that are (set-theoretically) supported $X$. (Category **2**)
> 
> 
> 


The usual reason I heard for why to consider the second category is that $\Diff(X)$ behaves badly when $X$ is singular, and specifically people will point out that $\Diff(X)$ is not Noetherian. (Noetherian = left + right.) For example, this is the case when $X$ is the 'cubic cone' [BGG72].


However, I am no longer satisfied with this answer because of the following:



> 
> (1), when $X$ is a curve then $\Diff(X)$ is Noetherian. [SS88]  
> 
> (2), when $X=V/G$ a quotient singularity then $\Diff(X)$ is Noetherian.
> 
> 
> 


But in these cases, one still considers Category **2** for these $X$. So it has to be the case that, in general and in these cases, $\Diff(X)$ is bad not just because it is not Noetherian, it is also bad for other reasons. So my question is:



> 
> **Question:** Why do we work in category **2** in the situations above. Or a better questions, what is **bad** about $\Diff(X)$ besides not being Noetherian.
> 
> 
> 


Note my question is not how to work in category **2**, but why it fails badly if we work in category **1** in situations (1) and (2).


It is worth to remark that:



> 
> in (1), if the curve is cuspidal then category **1** $\cong$ category **2**. [SS88] generalised in [BZN04]   
> 
> in (2), if the $X=\mathbb{C}^2/(\mathbb{Z}/2\mathbb{Z})$ then category **1** $\cong$ category **2** (I think this is true, but do please correct me if I am wrong.)
> 
> 
> 


[BGG72] I. N. Bernˇste ̆ın, I. M. Gel’fand, and S. I. Gel’fand. Differential operators on a cubic cone. Uspehi Mat. Nauk, 27(1(163)):185–190, 1972.


[BZN04] David Ben-Zvi and Thomas Nevins. Cusps and D-modules. Journal of the American Mathematical Society, 17.1:155–179, 2004


[SS88] S. P. Smith and J. T. Stafford. Differential operators on an affine curve. Proc. London Math. Soc. (3), 56(2):229–259, 1988.


  

**Noted later:** actually it is not true that on $X=\mathbb{C}^2/(\mathbb{Z}/2\mathbb{Z})$ then category **1** $\cong$ category **2**, sorry for the confusion.



Why is the ring of Grothendieck differential operators bad when $X$ is singular?
One thing that's wrong with the Grothendieck definition on singular varieties is the same thing that's wrong with defining the tangent space at a singular point rather than the tangent complex - it's not sufficiently derived (ie it's a strange truncated notion of the "true" derived object), which is why several different attempts to define D-modules give different answers there (the one quoted by OP, the notion of modules over vector fields - ie the algebra of differential operators generated by first order ones, which needn't be the full algebra in the singular case, and crystals - which amounts to the "option 2" one).


If you define D-modules in a derived fashion, as is done e.g. in the book of Gaitsgory-Rozenblyum, then order is restored: all the different natural notions you might come up with agree [here it's important to be in characteristic zero, a whole lot more interesting stuff happens in positive characteristic]. If you replace the naive notion of the sheaf vector fields by the tangent complex and consider modules for it, or define the Grothendieck differential operators derivedly (as a groupoid algebra for the de Rham groupoid of X), or define crystals (option 2), you get the same notion.


