Although it seems like the terms are often used interchangeably within the industry and marketing, there IS a technical difference between the terms:
A remake typically shares very little of the original assets and code with the original game, distinguishing it from an "enhanced port", partial remake, or remastering.
source: https://en.wikipedia.org/wiki/Video_game_remake
In essence, the difference comes down to just how much of the original game code and assets are used, with a "Remastered" title only updating graphics, whilst a "Remake" could, potentially, include radical changes in gameplay.
An example of a game that has had its graphics redesigned is Halo: Combat Evolved Anniversary,[11] while the core character and level information is exactly the same as in Halo: Combat Evolved.[11][12][13]
source: https://en.wikipedia.org/wiki/Remaster#Video_games
To use the Bethesda example, Skyrim with updated graphics only would be considered a Remaster, whilst (and good god I personally want this to happen) a version of Morrowind with both updated graphics and a Skyrim-esque combat system would be considered a Remake.
Hope that helps.