Texture art gives personality to 3D objects
Look the following model of a regular chair. With the current texture applied to it, you have no indication of who it belongs to, where it should be placed, or what is the context of the story it was included in.
However, once you create custom textures made especially for it, then you can really determine a lot about the owner, context and location. (Perhaps this chair is owned by a cartoon character, in a whimsical type of setting! The story is probably fun and cheerful!)
Textures indicate materials and add interest
Textures are also responsible for letting the viewer recognize the type of material that a specific object is made of. Besides identifying materials, textures also add detail and interest to otherwise plain models.
Textures disguise imperfections and finish off models
And finally, textures can greatly hide the imperfections of a poor model and can also help to add trim and finish off geometry that otherwise would look computerized and artificial.
Observe the two identical models below: the column on the left has a simple texture applied to it. The column on the right has texture that contains careful trims and decorative borders placed in strategic locations. Notice how the faceted polygons of the simple column are much more visible, while the intricate column's texture disguises it a bit. Also, the connection to the ground and ceiling are much nicer and finished on the column at the right.