First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
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 is definitely an open problem.
The only known cases are that of 0 and 1-motives. The case of (Artin) 0-motives is relatively easy, and the other one was proved through their equivalence with Deligne's 1-motives by Ayoub and Barbieri-Viale in their 2014 paper:
- Joseph Ayoub & Luca Barbieri-Viale, Nori 1-motives (2014)
For a recent reference (september 2015) of the general case being open, see section 0.2 of:
- Annette Huber & Stefan Muller-Stach, Periods and Nori motives (2015)
Example 3.20 of "Tannaka duality for enhanced triangulated categories", arXiv: 1309:0637v4 says that this would be true for rational motives if a t-structure compatible with Betti realisation exists. By Theorem 3.4 of Hanamura's "Mixed motives and algebraic cycles III" (MRL 1999), such a t-structure would follow from Grothendieck's standard conjectures, Murre's conjectures and a generalisation of the Beilinson-Soule vanishing conjecture.
In his thesis "Comparison of the Categories of Motives defined by Voevodsky and Nori (2016)" Daniel Harrer Compares "V. Voevodsky's geometric motives to the derived category of M. Nori's Abelian category of mixed motives by constructing a triangulated tensor functor between them."(compatible with the Betti realizations on both sides).