I really don't know what motives are, but perhaps any two smooth projective conics over $\mathbb{Q}$ have the same motive as $\mathbb{P}_1$. So you loose information about the places where the conic had bad reduction. The same applies to twisted forms of $\mathbb{P}_n$.

Also, perhaps any torsor under an abelian variety $A$ has the same motive as $A$, so you lose infomation about whether the torsor has a point or not.

Feel free to criticise or to correct me.

I really don't know what motives are, but perhaps any smooth projective conic over $\mathbb{Q}$ has the same motive as $\mathbb{P}_1$. So you loose information about the places where the conic had bad reduction. The same applies to twisted forms of $\mathbb{P}_n$.

Also, perhaps any torsor under an abelian variety $A$ has the same motive as $A$, so you lose infomation about whether the torsor has a point or not.

Feel free to criticise or to correct me.

Bonus. A bug in MO is preventing me from asking questions. While Anton gets it fixed, let me give a free translation of the above "answer" into French :

J'ignore complètement ce que sont les motifs, mais il me semble que toute conique lisse projective sur $\mathbf{Q}$ a le même motif que $\mathbf{P}_1$. Le motif ne se souvient donc pas des places où la conique avait mauvaise réduction. On peut en dire autant des formes tordues de $\mathbf{P}_n$.

Aussi, un torseur sous une variété abélienne $A$ a probablement le même motif que $A$; il ne se souvient donc pas si le torseur avait un point rationnel ou non.

N'hésitez pas à critiquer ou corriger cette réponse, ainsi que le français.

I really don't know what motives are, but perhaps any two smooth conics over $\mathbb{Q}$ have the same motive as $\mathbb{P}_1$. So you loose information about the places where the conic had bad reduction. The same applies to twisted forms of $\mathbb{P}_n$.

Also, perhaps any torsor under an abelian variety $A$ has the same motive as $A$, so you lose infomation about whether the torsor has a point or not.

Feel free to criticise or to correct me.

I really don't know what motives are, but perhaps any two smooth projective conics over $\mathbb{Q}$ have the same motive as $\mathbb{P}_1$. So you loose information about the places where the conic had bad reduction. The same applies to twisted forms of $\mathbb{P}_n$.

Also, perhaps any torsor under an abelian variety $A$ has the same motive as $A$, so you lose infomation about whether the torsor has a point or not.

Feel free to criticise or to correct me.