One can construct non-isomorphic smooth projective varieties which define the same motive by blowing up $\mathbb{P}^2$ at five points. I think I learned this here at MathOverflow. But these examples are all birational.
Are there any examples of non-birational smooth projective varieties which define the same motive?
More generally, how does one go about understanding the relation/difference between birational invariants and motivic invariants?
This is probably not the kind of example you are looking for, but you can get an example with Artin motives using rational coefficients and $\mathbb{Q}$ as the base field.
Let $K/\mathbb{Q}$ be a finite Galois extension with group $G$. Then $\mathrm{Spec}\,K$ is a smooth projective (not geometrically connected) variety over $\mathbb{Q}$, whose motive I will denote $M_K$. The subcategory of motives generated by $M_K$ is identified with the category of representations of $G$, where $M_K$ corresponds to the regular representation. For each subgroup $H\leq G$, the motive $M_{K^H}$ of the fixed field of $H$ corresponds to the permutation representation of $G$ on $G/H$.
Now, isomorphism classes of subfields of $K$ correspond to conjugacy classes of subgroups $H\leq G$, and isomorphism classes of representations of $G$ can be identified with class functions on $G$. So if there are more conjugacy classes of subgroups of $G$ than there are conjugacy classes of elements of $G$, there must be two distinct unions of subfields whose motives correspond to the same class function on $G$, i.e. whose motives are isomorphic.
We can get a concrete example with $G=S_3$. There are four conjugacy classes of subgroups of $G$, represented by the subgroups $\{1\}$, $\langle (12)\rangle$, $\langle (123)\rangle$, and $G$. Let $V_1,\ldots,V_4$ be the permutations representations corresponding to these subgroups, respectively. There are only three conjugacy classes in $G$, so there must be a relation among the characters of these representations. If I computed correctly, the relation is $$ V_1\oplus V_4\oplus V_4\cong V_2\oplus V_2\oplus V_3. $$ So e.g. taking $K=\mathbb{Q}(\sqrt[3]{2},\zeta_3)$ (with $\zeta_3$ a primitive third root of unity), the varieties $$ \mathrm{Spec}\, \mathbb{Q}(\sqrt[3]{2},\zeta_3)\sqcup \mathrm{Spec}\,\mathbb{Q}\sqcup \mathrm{Spec}\,\mathbb{Q} $$ and $$ \mathrm{Spec}\,\mathbb{Q}(\sqrt[3]{2})\sqcup \mathrm{Spec}\,\mathbb{Q}(\sqrt[3]{2})\sqcup \mathrm{Spec}\,\mathbb{Q}(\zeta_3) $$ have isomorphic motives. These varieties are smooth and projective, but not connected.
