Letting a person die is providing a person a release from death. It is as if he was suffering, and letting that person die is stopping his suffering, and letting him be at peace. He is no longer in the pain that death provides.
Bringing about a person's death, on the other hand, such as using a lethal injection on a person, is harming the person and putting him/her in pain. While in the other situation you are providing the person with relief, here, you are the one bringing about the person's pain.
But I believe that Rachel might think that letting a person die is much worse than causing the person pain because letting the person die if letting go of the person completely. You leave the person helpless, and as they are dying and screaming for your help, you just stand by and let it happen. You do nothing about it. You are simply a bystander, watching as said person slips into a wholly different reality altogether. Causing a person's death has a certain surety in it. You are causing a person's death, and you are probably doing it for a personal reason. There is a certain type of surety in causing a persons' death because you are the one causing it, while in the other situation, you are simply watching, letting it happen.
