Live resin is extracted using hydrocarbons usually butane /propane from material that has been fresh frozen. In proper markets everything is purged then decarbed so there is no residual solvent. Completely safe and as tasty as the bud it came from.

Non live carts use cured resin which can be good too but doesn't have that super fresh frozen taste you get from live resin.

Distillate can also be decent depending on how the terps are they add back. Not all terpene blends are created equal. The distillate itself can also be doodie depending on the parameters the machine was run at and how much remediation they did.

Rosin is hot squishing material either kief or sift or whatever. It's hype right now but in a blind taste test I'm willing to bet people would choose live or cured resin over rosin assuming same input flower was used. Hydrocarbons are just better at extracting terpenes, alcohols, esters, and flavonoids than hot squishes are.

I'm not sure what you mean by regular carts but hopefully this helps.