It was already stated that the problem is open, but I would like to add an important reference supporting the belief that the two approaches to motives are close (even if it might be terribly complicated to show they are equivalent).

The paper "An isomorphism of motivic Galois groups" by Utsav Choudhury and Martin Gallauer Alves de Souza (arXiv version of the paper) shows that Nori's motivic Galois group is isomorphic to Ayoub's motivic Galois group (which arises from the Betti realization of Voevodsky's motives). I guess (but I'm not quite sure about it) this says that if Voevodsky's motives could be shown to be the derived category of an abelian category, this abelian category would be equivalent to Nori's motives.

It should also be remarked that the t-structure (and hence the comparison) could only be expected for rational coefficients. Voevodsky showed that his motives with integral coefficients do not have a t-structure with the expected property, see e.g. the the discussion here.

It was already stated that the problem is open, but I would like to add an important reference supporting the belief that the two approaches to motives are close (even if it might be terribly complicated to show they are equivalent).

The paper "An isomorphism of motivic Galois groups" by Utsav Choudhury and Martin Gallauer Alves de Souza (arXiv version of the paper) shows that Nori's motivic Galois group is isomorphic to Ayoub's motivic Galois group (which arises from the Betti realization of Voevodsky's motives). I guess (but I'm not quite sure about it) this says that if Voevodsky's motives could be shown to be the derived category of an abelian category, this abelian category would be equivalent to Nori's motives.

It should also be remarked that the t-structure (and hence the comparison) could only be expected for rational coefficients. Voevodsky showed that his motives with integral coefficients do not have a t-structure with the expected property, see e.g. the the discussion here.