The fundamental difference in these approaches is that the agile approach is people-centric and the process maturity approach is process centric. In the agile approach, practices are introduced that are geared to supporting communication between people, making it easier for them to make changes to the software and minimizing the time that they need to spend doing things apart from software production (e.g. documentation).
The process maturity approach is based on defined processes that incorporate good practice and in ensuring that these processes are followed by all of the teams in an organization. They assume that by defining process and good practice, all engineers involved in development can perform in a comparable way. That is, they do not focus on the capabilities of the individual engineers but rather on being able to maintain consistent practice even although the team changes.