How does R-CNN and AlexNet compare?
I think you got most of it from the way you wrote your question.

How does R-CNN and AlexNet compare?
Are they used for the same purpose or R-CNN does more?

They are different things. AlexNet is a CNN architecture, i.e a neural network with a specific set of layers. R-CNN is multistep method that does object localization and classification using a search algorithm with a CNN (of any architecture) and a SVM.

Does R-CNN use Alexnet as a sub-module?

R-CNN can use AlexNet, or other CNN architectures, as its' feature extractor.