People use these two terms interchangeably. It can be argued that an project plan simply clarifies the resources needed for a project, and the action plan guides you through execution and control. The key point here is that having written plans that identify resources required, both financial and human, and a timeline for tasks to be completed, are critical for successful outcomes.
Some organizations use action plans to identify meeting schedules and topics as well.
However you incorporate planning, from the simple to the more extensive, you will appreciate the efficiency it adds.
Learning & Improving: Know How Non-profit
Project Management: Know How Non-profit
Section 5 - Developing an Action Plan: Community Tool Box
Basics of Action Planning (as part of strategic planning): Free Management Library
Action Plan for Board of Directors: The Denver Foundation
Tools & Publications: Community Resource Exchange