  In this paper we compare two constructions of weight functions (off-shell
Bethe vectors) for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$. The
first construction comes from the algebraic nested Bethe ansatz. The second one
is defined in terms of certain projections of products of Drinfeld currents. We
show that two constructions give the same result in tensor products of vector
representations of $U_q(\hat{\mathfrak{gl}}_N)$.
