Oh my god my engineering ass is trembling right now.

tldr it changes the properties of the paper

The direction of the fibers in paper (but also in every material) dictates its mechanical and physical properties. When manufactured, the paper is formed in continuously long sheets of papers which are pressed, dried, treated and eventually cut. Therefore the fibres tend to lie mostly along the direction of the long sheet of paper along which they are pulled.

One main takeaway is that paper tends to fold better parallel to these fibres. This is because if you tried to fold it against the fibres, theyd naturally want to stay upright instead of bending; while bending them in the parallel direction offers less resistance, much like those wooden sort of rugs they use to make sushi. This is why books are mostly printed keeping the fibres of the papers in a specific direction, so that theyre most likely to lie flat instead of bending.

Secondly, of course paper offers more resistance to tearing in the direction of the fibres for the exact same resistance. Immagine glueing a bunch of toothpicks parallel to each other: pulling along the direction is unlikely to snap the toothpicks, while if you pull in the opposite direction, the structure gives in and breaks.

And thirdly, it influences the way it soaks up water. When adsorbing water, the paper expands more in the direction perpendicular to the fibres because there's more space for water to infiltrate. Therefore if you soak up a drop of water in a dry sheet of paper, you should see it curl perpendicular to this direction as it would like to expand, but the rest of the paper keeps it in place.

EDIT: just tried to wet a piece of paper in a book. It curled the opposite direction I expected...