We revisit the competition between attractive and repulsive interparticle forces in simple fluids and how this governs and connects the macroscopic phase behavior and structural properties, as manifested in pair correlation functions. We focus on the asymptotic decay of the total correlation function h(r) which is, in turn, controlled by the form of the pair direct correlation function c(r). The decay of rh(r) to zero can be exponential (monotonic) if attraction dominates repulsion and exponentially damped oscillatory otherwise. The Fisher-Widom (FW) line separates the phase diagram into two regions characterized by the two different types of asymptotic decays. We show that there is a new and physically intuitive thermodynamic criterion which approximates well the actual FW line. This new criterion defines a line where the isothermal compressibility takes its ideal gas value χT=χTid. We test our hypothesis by considering four commonly used models for simple fluids. In all cases, the new criterion yields a line in the phase diagram that is close to the actual FW line for the thermodynamic state points that are most relevant. We also investigate (Widom) lines of maximal correlation length, emphasizing the importance of distinguishing between the true and Ornstein-Zernike correlation lengths.