Research proposals are written to justify a particular course of academic or scientific research, usually in a college setting. PhD students will use them, for example, when suggesting the topic they want to work on for their doctoral dissertation.
You might write a grant proposal if you wanted research funding but grants cover many other activities: low-cost housing, professional development, artistic projects and wilderness preservation, to name a few.
A research proposal identifies a problem, shows how the proposed research relates to it and places the research in the context of past studies and research in the academic literature.
A grant proposal states what the money is needed for, how the money is proposed to be used and why the individual or organization is the most qualified to receive the money. The exact form and requirements vary widely because there are so many organizations that provide grants.
What both proposals need are specific details: what the research will focus on; what objectives and goals will be accomplished with the money. This not only improves the chance of a good response, it will help the proposer focus on exactly what is needed and how to get it.