For all practical purposes, they mean the same thing. The best choice is "apply for". While "file for" can be used, and many people do use it, apply for is more direct. to file for something - means to make (fill out) an application for it I applied for a new job. I filed for a new job. These mean the same thing. I filled out an application. I filled out a file. The meaning of the second sentence is ambiguous. Is it an application or did you create a file folder for storing documents? Again, the sentence "I filed for a new job." is perfectly fine and well understood. It is just not used as frequently as "I applied for a new job.".

Here in America, one would say, "I filed my taxes." Seldom would one say, "I applied for my taxes.", as one noted difference between the two.