AW Michael's San Michel gains a +burst performance buff. In this instance it's a combination of +30% soft damage cap and +100% burst damage. Now keep in mind, what "+100% burst damage" means here is that it raises the multiplier by 1.00. A 3* SSR has a burst multipler of 5 and and awakened SSR's multiplier is 5.5. So "+100%" here raises them to 6 and 6.5, respectively. It's also the same type of buff as Shingen's PF (but different frame, so they would stack). Normally PF maxes out at +200% burst damage, but the HP weapon would raise that to +500%. Fortunately/unfortunately, there is a global cap on this specific type of buff, which is +500%. So that caps out an SSR/awakened SSR's burst multiplier at 10.0/10.5.

So what's the difference in burst damage between regular PF + San Michel++ and improved PF? Regular PF + San Michel++ adds up to a potential +300% burst damage; ie raising an SSR's burst multiplier from 5.0 to 8.0. If you have any 3* SR's in your team, their burst multiplier would be going from 3.5 to 6.5. Shingen herself with a 3* SSR weapon would go from 4.5 to 7.5. A 4*/FLB'd weapon should go from 5.0 to 8.0, I think?
Going back to improved PF, it's +500%. So 3.5->8.5 for 3* SR's, 5.0->10.0 for 3* SSR, 5.5->10.5 for awakened SSR, 4.5->9.5 for Shingen with 3* SSR, and 5.0->10.0 for Shingen with FLB'd weapon.
Then the next step to get the relative difference would be to look at the product of burst multiplier for one option multiplied by sum of elemental damage, then divide that by the product of burst multiplier for the other option and the corresponding sum of elemental damage, since there is a difference of up to +30% elemental there.

Without having done the actual math, my general expectation would be that the HP relic weapon should be better for squeezing out more spike damage, assuming you run a fair amount of +element to dilute the impact of that +30%. But also keep in mind that any additional sources of regular +burst damage (like Exceed) would start to close the gap between the two relic weapons, due to that global +500% burst damage cap.

Also, don't forget that you'll need to evaluate how important your damage output in between bursts is. If you're finding yourself placing more importance on spike damage and can get away with less DPS in the turns in between, then the HP relic weapon's strength looks more attractive.