I think it has to do with the amount of aluminum powder or whatever is used in the primer compound. If different primers were loaded into an empty case, so no bullet, no powder and fired from a 5" test barrel and a video or pictures were taken you would see a difference is obvious. So with some cartridges and powder combinations, you would have a difference as the above example I gave. Some cartridge and powder combinations will have little go no difference.
Here's another example, from Larry Gibson @castboolits,
"From a test I conducted sometime back with the 44 Magnum having primer pockets reamed for LR primers.
RCBS 44-250-K cast of WWs + 2 %tin with a BHN of 16, sized .430 and lubed with Javelina
WW Super 44 Magnum cases (new) with primer pockets reamed for LR primers
WW Super 44 Magnum cases (new) with standard primer pockets with control loads
OAL; 1.711"
Powders; Alliant 2400 (21 gr) & H110 (23 gr)
Results listed as; primer/average velocity (fps)/SD/ES/average PSI (M43)/SD/ES
H110 load;
Control load;
WLP/1400/24/78/21,700/1,200/3,600
LR primer loads;
Rem 9 1/2/1466/47/112/25,200/3,500/8,500
CCI 200/1390/49/141/22,100/1,500/4,200
CCI #34/1507/35/107/28,300/3,000/9,000
WLR/1506/26/85/28,500/2,300/6,000
Fed 215/1544/12/36/31,600/1,500/4,100
Here we see all the psi's within SAAMI proscribed MAP. We find the control load with WLP primers to be relatively mild. The hotter Fed 215 magnum LR primer produced the most consistent internal ballistics and the highest psi with the highest velocity with H110. No pressure problems observed with using the LR primers in this load with H110 powder.
Alliant 2400 load;
Control load;
Fed 150/1542/18/51/30,800/1,000/2,700
LR primer loads;
Rem 9 1/2/1603/13/37/34,700/500/1,200
CCI 200/1589/20/46/33,000/1,800/5,000
CCI #34/1634/10/27/37,900/1,900/5,500
WLR/1633/8/22/37,600/1,000/2,600
Fed 215/1641/13/29/41,200/1,900/4,100
Here we see some LR primers (the "soft" 9 ½ & 200) are OK with this load keeping the psi under the SAAMI MAP/. However, when magnum level LR primers are used we see the psi going Over the MAP and the Fed 215 psi going over the MPSM. Note the very good internal ballistics with this load of 2400 with all the primers tested. Obviously, this load is burning very efficiently in the 30K to 41K psi range."
So, a cartridge like a 9mm or 45 ACP, you may not see a difference through your chronograph or if you had pressure test equipment, no matter what primer you use. Even a larger case like a 45 colt or 44 magnum loaded with a faster powder, or even a certain slow powder, no difference.