Short Version: Gore-Tex and eVent both use a PTFE waterproof membrane. eVent is generally more breathable than Gore-Tex but cooler to wear and requires more regular washing and treatment to maintain performance and lifespan.
Long version: Gore-Tex and eVent are both waterproof fabrics used in outdoor clothing. Gore-Tex and eVent are actually names for the waterproof membrane which makes up just one layer of the waterproof fabric. The membranes can be used in double or triple layers and are bonded to an outer face fabric for durability and possibly a inner fabric or scrim. Both Gore-Tex and eVent use the same PTFE membrane to repel water and let out sweat. It's the bit everyone describes with the holes too small for water droplets to pass but large enough for water vapour to pass. So rain is held out whilst sweat vapour can escape. The problem with PTFE membrane is that it is very fragile and easily contaminated by dirt, skin oil and other chemicals present on your skin. Without some protection a thin layer of PTFE would not last long. So the difference between Gore-Tex and eVent is how they protect this PTFE membrane.
Gore-Tex protects the PTFE with a layer of PU coating. Now PU coating is actually used on it's own to create much lower performing waterproof fabrics that use no membrane, however when it is used to protect the PTFE it is able to be used in a much thinner layer than would not be waterproof on it's own but is enough to protect the PTFE membrane whilst retaining breathability. The crucial effect of the PU layer is that water vapour (your sweat) must condense on it, pass through as a liquid, and re-evaporate on the other side to pass through the PTFE. Whilst this provides excellent protection for the membrane, it does reduce the breathability of the jacket. As mentioned above the number of layers and face fabric also affect breathability, so which Gore-tex you have is also relevant. 3 Layer Proshell with Ascendor fabric is one of the most breathable combinations which gets close to eVent in breathability where as Gore-tex Paclite is nowhere near.
eVent also protects the PTFE with a PU coating but rather than the PU being a continuous layer of PU between the user and the PTFE, high pressure air is used to drive the PU INTO the holes in the PTFE so rather than being a layer in front of the PTFE the PU has coated entire surface area of the PTFE but does not sit continuously in front of it. Now there is no need for the water vapour to condense and re-evaporate as it can pass directly through this version of the PTFE, what eVent call Direct Venting. However using the PU in this way does not do such a good job of protecting the PTFE from contaminates and means than the fabric requires regular cleaning to maintain performance. There is another consequence to Direct Venting and that is that the heat inside your jacket also has an easier path to the outside. Yoy can see this as both a good and bad thing depending on the situation but just remember than an eVent jacket is cooler and less insulating than a Gore-Tex jacket so you may need to adjust your inner layers.
Summary: Some users are not really bothered which fabric is used to waterproof their jacket so long as it works suffciently, which both Gore-Tex and eVent do. However a waterproof garment is a big purchase and it's nice to know what performance you will get and what maintenance you will be required to do over the life of your garment. There is no definitive answer to which one is best, so much depends on who you are and what you do, that's why it's nice to have the choice.
Below is a short video showing the direct venting process in action. Whilst this is impressive, it must be remembered that venting zips, open sleeve cuffs, pitzips, unzipping your jacket and the holy grail of taking the thing off combined with the correct layers underneath all play a much greater part in keep you dry, comfortable and performing maximally than the difference in breathability between waterproof fabrics.