TL;DR: remaster=close to original, remake=close or far from original, depending.
These words are meant as analogies to the equivalent in music and films.
In music, a remaster specifically means to put together a new "master copy", a recording from which all CDs, Tidal tracks etc will be made. When a song was produced, the vocals and instruments may all have been stored on different tapes, before being edited and processed into one single "master". remastering redo this job.
A "remake" on the other hand, is a new and independent recording of a film, based on an earlier film. When this happens to a brand, such as star-trek, we talk about a "reboot".
In games, a typical example of a remaster would be the new releases of Monkey Island. Gameplay-wise they are exact replicas of the old games, so much that you can switch between new and classic modes at a keypress. A "port" of a game to another platform would also be a remaster.
An example of a remake would be Prince of Persia, where the basic concept of a jumping and slashing guy dressed as lawrence of arabia was kept, but the storyline and gameplay otherwise altered.
So.. Crystal clear? Unfortunately not. :-) These words are just analogies that people do with as they please. On wikipedia the article Video game remake use "remake" as I've used "remastered" here and use "reboot" for what I've called a remake. (Being part of a franchise is why you remake old stuff, hence the "reboot"="remake")
You can see the confusion in that Wikipedia has a List of video game remakes and these include several games with the word "remastered" in them, such as "Day of The Tentacle - Remastered" and "the last of us remastered".