That is by far the most popular question I get asked by promising new Art Department candidates. I think it depends on each individual person and their situation. I don't have an answer. However, I can tell you I wish I had gone to Film School.
Film School for filmmakers is like Art School for artists. You get to explore the multi disciplines and mediums that fall under the profession. You don't need to have a degree to be a successful filmmaker, or artist. But a solid background is key. You need some basic design and filmmaking experience and/or education. Next best thing to Film School is volunteering on indies. You get to do it all.
Because I fell into the factory production end of filmmaking, and arrived with a narrow view, I learned fairly late in my career that I was also equally, if not more passionate about working in other departments. I wish I had discovered my other passions earlier. Not only that, I think that by working in other departments, it made me a better designer, as I understood the creative process from different perspectives.
Subjects that really helped my Art Department career were Drafting, Graphic Design, Illustration, Architecture, Interior Design, Art History, VFX and Math. I use those disciplines all the time, and I think most of my associates do too. The more disciplines you can master, the more indispensable you will be.