Project management maturity is a measure of two factors:
Consistency of practice.
Complexity of practice.
Project management maturity applies to an organization, such as an IT department or a research lab. The concept doesn't apply to people.
These two variables are pretty easy understand:
Consistency of practice means that from project to project, people are using the same techniques, following the same guidelines, and applying the same heuristics. Each project manager produces plans and reports that follow the same format and have the same kind of information. As a result, the organization develops habits and the organization can evaluate whether the habits are effective or not. The other benefit is that future project leaders grow up in an environment where good practices are all around them.
As simple and practical as this sounds, many organizations still haven't picked up on it. Instead of establishing common standards and teaching people to follow the standards, they prefer to focus only on "getting project managers certified." This theory holds that if everybody has read the same book they'll all have the skills to apply the theory. But a bunch of good project managers all doing it their own way is not consistent practice. And it doesn't build organizational habits. So the individual project leaders learn and improve, but overall, the organization hasn't learned anything.
Complexity of practice recognizes that project management is a big field with a lot of proven techniques. A firm can get value from doing the basics of project management consistently. For example, every project has a sponsor, a project manager, a statement of goals and scope, and an action plan with tasks, due dates, and responsibilities. That is a starting point. Now add that every project generates a risk plan and keeps it up to date. That is a little more complex. If all projects are doing the basics, then they all add risk management, the organization has matured.
Increase Project Management Maturity by repeating these steps:
Train people to be able to follow the standards.
Leverage technology as appropriate.
Coach people to improve their ability to meet the standards.
Hold leaders and teams accountable to using the standards.
Repeat the process with more complex standards.