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.
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, while the core character and level information is exactly the same as in Halo: Combat Evolved.
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.
UPDATE: Having asked a friend who used to work within the industry (on several remastered titles), he confirms this is the way it is supposed to be used. He also said, however, that even within development and testing the words were often used interchangeably both verbally and on documents.