Is there any significance to a weapon's weight?

I am currently in the first part of NieR: Replicant for the PS4, and I have obtained a number of one-handed swords. Each sword has its own unique stats, story, length, and weight.

As far as weight goes, there are various stats ranging from very light to heavy. I have seen slight differences in attack range while wielding longer weapons, yet I have not seen any differences when using weapons of different weights. I would have expected this to affect combo speed in battle, or movement speed while traveling the overworld, but this does not seem to the case as far as I can tell.

What is the significance to a weapon's weight, if any at all?

There actually is significance, though not much. A lighter weapon will swing/strike faster than a heavier weapon within the same weapon class (1H, 2H, and spears). Might be worth mentioning that in the first part of the game, you will only have access to the one-handed swords.

Naturally, spears and one-handed swords will be faster than two-handed swords. You can also augment your attack speed (and more or less cancel out the effect of weapon weight) by equipping Physical Strength+ words wherever possible.

From a GameSpot forum post:

Heavier weapons strike more slowly but this can easily be mitigated by equipping +physical strength words to your martial arts, so it doesn't really matter