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.
The only difference with the student version is who can buy it.
And it's not just "student" - it's Teachers and Students. And really, just about anybody under the education umbrella, depending on how much work you care to do.
There are some good articles out there (including Adobe's website) that describe who can qualify with proof (which varies based on type).
For regular college students, simply using a .edu address is enough.
Regarding license, it's a full license with no commercial restrictions. You can do anything and everything that you have the right to do with a standard retail license.