Josh - you're absolutely right in recognizing these equations are closely related; if we plug them into one another (and keep our +/- signs consistent!), we see how the whole thing collapses into something we already know.
Remember that in ΔE = q + w, when the system does work on the surroundings (expands) we consider this negative, so w is equal to (-PΔV).
When we put that into our enthalpy equation ΔH = ΔE + PΔV, we see that ΔH = (q -PΔV) + PΔV, or ΔH = q. The mechanical work term completely drops out of the equation.
This means that a change in enthalpy depends on the heat input.