The FE exam is to become a licensed professional engineer. It is certainly not required to be considered an engineer, and is considered a waste of time if you are doing work that does not require it. The license is most effective for civil engineers, because building/bridge/etc designs need to be approved by a licensed engineer for regulatory purposes. It was pretty common for mechanicals to take the exam too. I assume it would be beneficial to get it if you are doing any sort of government related work.

I studied computer engineering and have no desire to take the FE exam, as being licensed is almost entirely useless in my field. As far as the difference between "engineers" and CS majors, at my school basically the only difference is math requirements, and how math is used in the higher level classes. In ECPE (computer/electrical) we differentiate ourselves from CS by studying circuits, systems, physics which are very calculus-intensive. The area where it crosses over into CS is when you get into computer architecture, digital design, assembly, which is like programming, but due to being so close to the hardware level (working with raw digital logic) it ends up being more math intensive than most high level programming.