I'm applying to some schools in US for my PhD studies. In the application form there are two different places for uploading CV and Resume.
What is the difference between them?
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A resume tends to be a one page summary of what your background is with respect to the position you are applying for. informally needs to be a very dumbed-down version of what you have done so it can be understood by a general audience specially the early-stage selection committees(read departments staff, random volunteers, HR personnel). In a resume, a person might focus on marketing his accomplishments in a context of the positions this person is submitting the resume to.
CV or curriculum vitae can go any number of pages and can include all aspects of you. I have heard people having 150 pages of CV and using them for applying to positions like a program manager at NIH with 40 pages for their publications alone. A CV is in a way an absolute document and you don't edit it based on the position you submit your CV to. It is generally all there is to it.
It may be worth pointing out that the distinction made between resumes and CVs in the other answers is very much a US thing. Outside of the US, things are different. In Europe (including the UK) resumes are uncommon. Instead, CVs are used for everything. However, CVs are generally expected to be much shorter. As a rule of thumb, a European should fit on two pages A4, and is tailored to the job you are applying for. Also, unless specifically requested, a European CV will not contain a list of publications. European CVs thus fall somewhere in the middle ground of American resumes and CVs.