  We study the $q\bar{q}$ potential in strongly coupled non-conformal field
theories with a non-trivial renormalization group flow via holography. We focus
on the properties of this potential at an inter-quark separation $L$ large
compared to the characteristic scale of the field theory. These are determined
by the leading order IR physics plus a series of corrections, sensitive to the
properties of the RG-flow. To determine those corrections, we propose a general
method applying holographic Wilsonian renormalization to a dual string. We
apply this method to examine in detail two sets of examples, $3+1$-dimensional
theories with an RG flow ending in an IR fixed point; and theories that are
confining in the IR, in particular, the Witten QCD and Klebanov-Strassler
models. In both cases, we find corrections with a universal dependence on the
inter-quark separation. When there is an IR fixed point, that correction decays
as a power $\sim 1/L^4$. We explain that dependence in terms of a double-trace
deformation in a one-dimensional defect theory. For a confining theory, the
decay is exponential $\sim e^{-ML}$, with $M$ a scale of the order of the
glueball mass. We interpret this correction using an effective flux tube
description as produced by a background internal mode excitation induced by
sources localized at the endpoints of the flux tube. We discuss how these
results could be confronted with lattice QCD data to test whether the
description of confinement via the gauge/gravity is qualitatively correct.
