I am a brand new member of this forum, but not a brand new brewer, and I have experience with a 10 gallon cooler all-grain setup with both a braid, a mesh "bazooka tube" screen and, later, a copper manifold using both batch sparging and fly sparging. For the sake of the conversation, I will pretend the braid and the bazooka screen are the same thing, which they pretty much are in terms of lautering results.
As stpug wrote, the screen versus manifold method of lautering should not make a difference in batch sparging aside from the differences in dead space in the lauter tun between the two setups. It makes sense that given this, a brewer would experience a loss in overall efficiency by leaving more sugar water in the lauter tun with the braid than with the manifold. You can quantify this difference by calculating the difference in dead space between the two methods. Even so, if you are not no-sparging and leaving a ton of dead space with the braid, the small amount of weak wort kept out of the boil kettle by the additional dead space of the mesh screen method should not impact efficiency more than a percentage point or two. The more batches in your batch sparge, the less the difference matters.
When continuous sparging, there will be efficiency differences due to preferential flow in the grain bed and certain portions of the mash being exposed to more rinsing than others by the sparge water. As others have written, Palmer covers this pretty extensively, and for continuous/fly sparging the rule of thumb is that a false bottom is most efficient, followed by a well designed slotted copper pipe manifold, followed by a mesh screen or braid, with a single point of collection being the worst.
When I switched from a tubular screen to a copper manifold my efficiency increased by 5+ points. I now employ a recirculating mash and fly sparge using a copper pipe manifold and I am very happy with a typical efficiency of 83%-85%. Theoretically, with a false bottom I could do even better. BTW, I abandoned the braid early on. For a number of reasons, it just didn't seem to work as well in my setup. I have stuck with the manifold and even made a larger one rather than go to a false bottom when I designed my "new and improved" rig. YMMV...