Both eVent vs GORE-TEX achieve their waterproofing in a similar way; they both start with a Teflon material PTFE which they then extrude and stretch until pores form in the material and the pores are microscopic that inhibit water from coming in but still allow air and water vapor to escape.
Where the difference between eVent and GORE-TEX is in how they protect the membrane from skin oil and from pollution.
GORE-TEX, a wet system, came up with a 2-stage venting process, by putting a PU layer on the top and bottom of the membrane, so the PU layer protects the membrane from pollution but at the same time, it covers all these tiny pores, making the material lose its air permeability. The PU is thin enough to allow moisture to pass through but not air.
eVent is a dry system, which starts with the same PTFE membrane, but instead of putting a layer of PU protectant, they use a chemical coding that coats the individual fibers of the membrane on a microscopic level leaving the pores intact. Instead of having to get this PU layer wet before the garment starts to breathe, the moisture and air both can move through the membrane to the outside and evaporate. It's a much faster and efficient venting process.
Check out a demonstration of air permeability in this video (5.52):