Let
g
\mathfrak {g}
be a complex, semisimple Lie algebra, and
Y
ℏ
(
g
)
Y_\hbar (\mathfrak {g})
and
U
q
(
L
g
)
U_q(L\mathfrak {g})
the Yangian and quantum loop algebra of
g
\mathfrak {g}
. Assuming that
ℏ
\hbar
is not a rational number and that
q
=
e
π
i
ℏ
q= e^{\pi i\hbar }
, we construct an equivalence between the finite-dimensional representations of
U
q
(
L
g
)
U_q(L\mathfrak {g})
and an explicit subcategory of those of
Y
ℏ
(
g
)
Y_\hbar (\mathfrak {g})
defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian, additive difference equations defined by the commuting fields of
Y
ℏ
(
g
)
Y_\hbar (\mathfrak {g})
. Our results are compatible with
q
q
-characters, and apply more generally to a symmetrizable Kac-Moody algebra
g
\mathfrak {g}
, in particular to affine Yangians and quantum toroïdal algebras. In this generality, they yield an equivalence between the representations of
Y
ℏ
(
g
)
Y_\hbar (\mathfrak {g})
and
U
q
(
L
g
)
U_q(L\mathfrak {g})
whose restriction to
g
\mathfrak {g}
and
U
q
g
U_q\mathfrak {g}
, respectively, are integrable and in category
O
\mathcal {O}
.