CCI military sensitivity spec (less sensitive than commercial) #34 and #41 primers do not have thicker cups. CCI told me over the phone those primers have cups and priming compound formula and quantity identical to CCI 250 and 450 magnum rifle primers, respectively. The lower sensitivity is achieved by using an anvil whose legs have a wider included angle of spread, making them a little shorter and a little less rigid than the anvils they use in the 250's and 450's. So these are really a modified commercial primer. Actual mil-spec primers go through a lot of additional lot sample testing and are a lot more expensive, therefore. The only military sensitivity spec primer I am aware of that uses a thicker cup to reduce sensitivity is the Federal GM205MAR for 5.56 (they don't make a large rifle military sensitivity spec primer). Unlike the CCI #41, it is not a magnum primer, but rather is their 205M match primer in the thicker cup.
The CCI BR primers are, as mentioned before, simply a standard primer design executed to higher QC requirements, with selected cups and anvils and the priming mix dosage introduced with an eye to higher consistency by their most experienced primer making personnel.
CCI claims on their site:
An independent researcher identified the use of CCI Benchrest primers as one of two factors that were the most significant contributors to tiny groups.
It may be hype, but if they are genuinely more consistent it probably isn't. It's just that for reasons of how standard deviations add up, it won't be apparent unless you are trying to tighten groups of a quarter moa or less. Also, in every situation where the combination of powder and bullet are different, you can get different results. I've read recently someone saying BR primers did the worst in his experiment. There are a lot of variables here that may not be the primer's fault, though, like different seating effort messing up the handloader's use of them, or slightly lower sensitivity than, say, a Federal primer causing a mainspring that has taken a set to fail to strike with enough energy to make a less sensitive primer fire consistently.
If your gun is not shooting under 1 moa, the component combination typically matters less than if you are, but as with all other rules about loading for accuracy I know of at least one published exception where groups went from 2 moa to 1 moa with one change to one component (outside neck turning of the brass) and have personally had flash hole deburring take loads of one powder from about 1.25 moa to 0.7 moa (but I could easily get that gun to shoot 0.7 moa with other powders without resorting to deburring).
Until you try something in your gun, nothing is certain about it.