Differences between cover letter and research plan?
In general, the discussion of research in a cover letter is much briefer than a research plan. The former is generally one or two paragraphs, while the research plan for PhD positions is typically one to two pages.

A good "research plan" (or in this case, more accurately a "statement of purpose") should explain why and how your past and present experiences prepare you to undertake the topic you'd like to study as a PhD student. Note that this is not meant to be a "life story"—we don't care what you did when you were five years old. Instead, tell us about relevant experience: research, professional experience, service work, and so on.