[AndyJackson]

It isn’t the same. Leave out the relative clause “who is without sin” and just keep the main clause.

The choice is between “Let he throw the first stone” or “Let him throw the first stone.” “Let him” is obviously the grammatically correct choice. That is because him is the direct object of let.

[Lagmeister]

I think it is one of those subject understood - as in:

(You)let him (he), who is without sin, cast the first stone.

[mkmensinger]

I will defer to those who say “him”, although the grammarians I checked differed from each other. I do think we might be picking some nits here.