The new series is not just a re-make. The basic cast and situation is preserved, but that is about where it ends. The entire serial was given a makeover as a serious drama, and the characters fleshed out into real people with complex relationships. In fact, almost 50% of the story is dedicated to the interactions between various family members, and the other to creative world building, all to great effect. There are story arcs that were thought out from the beginning. It plays like a Robinson Caruso in space, which I believe was the original intention behind the show in the 1960's.
On that note, after the first season I researched the original show a bit further and found that Season 1, which was in black and white and never seen in my youth, was a much more serious drama that actually parallels much of the original story line elements in the new series. You can see in the original pilot episode the remarkable elements of 60's sci-fi era films. In the new series, the changes they make are also signs of the times but well integrated. The mother Maureen is the strong scientific genius of the family, and Dr. Smith is now a woman, although just as clever and shifty as in the the original pilot (Dr. Smith becomes something of a caricature in later seasons of the original series - "Oh the pain, the pain!"). And the robot and Will still share a bond but this robot is a lot cooler and is actually part of an alien race. The only thing he still says from the old series is "Danger, Will Robinson". Just enough tribute without the cheese.