There is no difference in capability. Its about the license restrictions.
Usually student versions are restricted to personal use and can't be used professionally. I.e. if you decide to do photography for money you can't use the student version to process the photos that you plan to sell.
yes, the software is the same, just licensing concern. anyone enrolled in any accredited university (online included) or still in high school is eligible for the student version.
And, who is a student?