This "Spider Wasp" has never killed any spider, because this is a (male) specimen of Sphex pensylvanicus, a Katydid hunter of the Sphecid, not Pompilid, family.
One tip: true Spider Wasps never have such a stalked gaster.
As for the spider, this one of the famous "Crab Spiders" (family Thomisidae). These are among the worst enemies of Hymenopterans, easily subduing even female, stinger-armed individuals (including a great plenty of Honeybee workers).

By "stalked gaster," you mean the narrow tube joining its abdomen to its thorax, correct? If so, you're quite right; in my observational studies, I have noticed that such stalked gastra are indeed a hallmark of certain species within family Sphecidae.

This "narrow tube" (technically, the petiolus) consists mainly of the modified 1st urosternite, i.e., first ventral gastral segment.
Although quite variable in length, this structure can be found in all members of the current family Sphecidae.
Among former traditional Sphecidae, now placed in family Crabronidae, most members of subfamily Pemphredoninae have a similar one.
But all Pompilidae have a sessile (i.e, without any true petiolus) gaster.

After all these years, I now, finally, know the proper terminology to describe the aforementioned 'hallmark' of Sphecidae!