A proposal format for that class project requires student to provide research plan as well as proposed approach. Can you please help me in understanding differences between the two?
In general a research plan will include within it a section that deals with the proposed approach. As this University of Michigan page explains:
Research Plan (Approach)
This section includes a comprehensive explanation of the proposed research, and is addressed to other specialists in your field (not to laymen). The section is the heart of the proposal and is the primary concern of the technical reviewers. To make it clear and easy to follow, you may need several subsections tailored to your work. Research design is a large subject and cannot be covered here, but a few reminders concerning frequently mishandled aspects of proposals may be helpful.
As always, you should talk with your advisor/professor to ensure if this is exactly what they have in mind, or if there is some specific template or example for your field/subject/use that you should follow.