I’m not familiar with that plugin. Based on its description, there is a lot of functional overlap. From what I can see:
- The Groups premium extensions provide full e-commerce integration and some other add-ons which I have not pursued.
- Groups emphasizes group hierarchy, where Press Permit relies instead on multiple group memberships.
- Press Permit Pro customizes content creation and editing permissions.
- Press Permit can customize access per-category. Not sure if Groups does that.
- Press Permit customizes permissions for existing WP role groups, custom groups or individual users.
Although I have not tried out Groups, it appears to prioritize simplicity, polish and drop-in solutions for membership sites. With Press Permit I also aim to be clean and friendly, but tend to focus most on versatility (especially for collaborative editing with Pro). It seems to me that both plugins will continue to improve as the authors take some clues from each other – while maintaining distinct approaches to fit their target user base.