For example, I am looking forward to the new Grim Fandango, but it is the SAME game. Should I move on and enjoy the newer version or just play the old version that I have?
Not all digital 'enhanced' remakes are the same. Some are absolutely terrible. I don't know where Grim Fandango will fall on this list. The remakes for Monkey Island 1 and 2 were pretty good, but part of that is the ability to switch in and out of the old and new graphics at any time. So the best of both versions.
Gold Rush remake has terrible voice acting, the graphics look like 1995-1998 quality, although technically HD. The animation is stiff. IMo, the original is better.
I don't consider the new Gabriel Knight 20th Anniversary to be an improvement over the original. The older one had better actors, better atmosphere, and better/more animation even though it was 2D. I prefer the original over the new one, although the new one is still fun.
The Leisure Suit Larry Reloaded isn't necessarily better than the original or the remake.
The Police Quest 1 remake isn't necessarily better than the AGI version, etc.
KQ1 AGI and KQ1 remake, and the fan remake? They each have their pluses and minuses, and reasons to play them.Puzzle differences between the original and the remake for example.
The New Shadowgate, now was an exception. It was a lot of fun, original puzzles, and great atmosphere. Both versions are enjoyable for their own merits.