The boto package is the hand-coded Python library that has been around since 2006. It is very popular and is fully supported by AWS but because it is hand-coded and there are so many services available (with more appearing all the time) it is difficult to maintain.
So, boto3 is a new version of the boto library based on botocore. All of the low-level interfaces to AWS are driven from JSON service descriptions that are generated automatically from the canonical descriptions of the services. So, the interfaces are always correct and always up to date. There is a resource layer on top of the client-layer that provides a nicer, more Pythonic interface.
The boto3 library is being actively developed by AWS and is the one I would recommend people use if they are starting new development.
