What is a freelance writer?
A freelance writer is not under contract to work for a magazine or newspaper, and can sell their work to the highest bidder. From what I understand most have an area of expertise that they specialize in. While it is possible to do freelance work as a university student, it may not be the most steady source of income. I think that it is a tough over all way to earn money, especially if you are new at it. However, it is all clear that some writers do earn their living this way, so it is not impossible.
What is the difference between grant writing and proposal writing?
A grant writer is asking for something. A proposal writer is offering something. While they are different, the "fundamentals" of writing them are the same. A grant is written asking for funding for a project. It includes information about your organization and a description of what you plan to do with the money should you be given it. Grant writing is generally a function of non-profit charitable organizations. A proposal is written suggesting your company will provide some service to an individal or other company. It includes information about your organization, what you intend to offer and what you expect in exchange. Proposal writing is usually a function of private sector businesses.