TITLE: Does a motive capture everything about an algebraic variety?
QUESTION [6 upvotes]: Is the functor from the category of projective varieties over a field $k$ to the category of pure motives over $k$, faithful? (Perhaps it is not full).
Ditto:
Is the functor from the category of affine varieties over a field $k$ to the category of mixed motives over $k$, faithful? (Perhaps it is not full).

REPLY [2 votes]: If you work with Grothendieck motives modulo rational equivalence, you even have simpler examples : the motive of any split (projective) quadric of odd dimension $n$ is the same as the motive of $\mathbb{P}^n$.