Hi David,
The Genesis connector is the legacy connector (developed even before Campaign became part of Adobe) and is more similar to a direct integration with Adobe Analytics, while the People core service is more similar to integration with Audience Manager, and became available only a couple of years ago after Adobe started its Marketing/Experience Cloud.
I don't really know the pros and cons for each, just wanted to highlight these facts to explain why they are both available, in case it can be useful for you. People core service/Audience Manager is the most recent solution.
As an advice, I would say go for People core service if you plan on leveraging more Adobe solutions in the future, to have this service centralize your data.
Hope this helps,
Florent