Just to add to what bemdesign said, wordpress premium is a service that you get on top of the wordpress.com service (adds features -onlocks a few more of them- to the free wordpress.com services), www.remarpro.com is software that you can install on a web server.
If you just want to install a wordpress on a webserver, and *actually* don’t need any of the additional abilities that www.remarpro.com makes available then you could just go with Premium. But you would not need both of them, ever. Because “premium” would only be available on wordpress.com, not on a self-hosted www.remarpro.com.
as far as features: with www.remarpro.com
Depending on the host the storage space would vary, and depending on the theme you chose your design options would vary, and the only ads you got would be ones you put there (that are paying you rather than paying wordpress.com).
wordpress.com – premium eliminates wordpress.com’s ads, “videopress” replaces video plugins that you can install on www.remarpro.com, “custom design” replaces the way some themes have additional customization features in their themes dashboard widgets, the extra space that you get (13 gigs) is relative to however much space you sign up for with the host that you pick (typically all hosted accounts are going to have more space than that anyway – bluehost would start at 100 gigs and go up from there), to have a hosted account you would get a domain name through the host.
You would just need to look at both and see which one works for you.