In this paper we study the relation between the Lefschetz number and the Nielsen number of an Anosov diffeomorphism on a flat manifold. As a first result we obtain that for each n >= 4 and each k satisfying 2 <= k <= n - 2, there exists a flat n-dimensional manifold M having first Betti number b(1) (M) = k and admitting an Anosov diffeomorphism f on M with N(f) not equal vertical bar L(f)vertical bar. On the other hand, in almost all cases one can also construct on the same manifold M an Anosov diffeomorphism g with N(g) = vertical bar L(g)vertical bar. Analogous results are obtained in the case of primitive flat manifolds M, i.e. with b(1) (M) = 0. Since flat manifolds with b(1) (M) = 1 or b(1) (M) = n - 1 admit no Anosov diffeomorphisms, and the class of flat manifolds with b(1) (M) = n consists entirely of tori, a complete picture is obtained.