They're not the same. 'Train spotter' is a cover all term used by the general public and the mass media...even if number-taking isn't your bag at all.
People like to taking photos, making models (garden railways/layouts/live steam/model engineering), researching history, travelling over different lines or line bashing if you will, haulage bashing, volunteering on heritage railways, collecting memorabilia...all this can't be pigeon-holed into 'Train spotting'.