TITLE: why are motives more serious than "naive" motives?
QUESTION [15 upvotes]: I know my question is a bit vague, sorry for this.
Let $k$ be a field of characteristic zero. Consider the Grothendieck ring of varieties over $k$, usually denoted by $K_0(Var_k)$. This is generated by isomorphism classes of varieties over $k$ modulo the relations [X]=[Y]+[X-Y] whenever $Y$ is a closed subvariety of $X$. People usually refer to [X] as the "naive" motive of $X$. 
On the other hand, one has Voevodsky's "true" motives $DM_{gm}(k)$ (not as true as we would like to, I know !) and to any variety $X$ we can attach an object $M(X)$in $DM_{gm}(k)$. 
Why is this $M(X)$ more serious than the naive one? That is, can you give some examples of properties that cannot be read at the level of $K_0(Var_k)$ but that one sees when working in $DM_{gm}(k)$?

REPLY [10 votes]: Note that the Grothendieck ring of varieties does, at least conjecturally, remember some information about varieties that the category of motives does not.
Under the cut-and-paste conjecture, two varieties are equivalent in the Grothendieck ring if and only if they can both be decomposed into the same set of locally closed pieces. There are many pairs of varieties which have the same motive but cannot be cut and pasted into each other (like a $\mathbb P^2$ and a fake projective plane). So assuming the cut-and-paste conjecture, the Grothendieck group remembers a lot of extra information about the varieties.
In particular, because the fundamental group is a birational invariant for smooth projective varieties, the Grothendieck group class tells you the fundamental group, which is highly non-abelian invariant.
So the category of motives loses some information that is contained the Grothendieck group of varieties - in some sense, the non-abelian information. One reason that the category of motives is good to work with is that it simplifies things by getting rid of this additional structure. For tasks where you only need that structure, motives are more serious.