In order to achieve the effect you're looking for, you need to apply your texture in such a way that affects the displacement 3D structure of your base model, or at the very least gives the appearance of displacement. In 3DS Max, you either want to use your texture as a bump map or displacement map for your model. I will explain what each of these techniques do in this post--for an explanation of how exactly to get each of these effects in 3DS Max, see this tutorial.
A displacement map will actually physically move the points on your base model up or down, ultimately affecting lighting, shading, and the overall appearance of your model to give it a realistic textured look. This can slow down your rendering times, but it gives by far the most realistic results for the effect you're looking for.
Another option is to use a bump map, which mimics the lighting and shading effects of a displacement map, but without actually moving any parts of the model. The explanation behind bump mapping is a little technical, so I won't get into it here, but just know that it's really useful if you need to speed up your rendering times while still achieving a 3D textured look.
As you noted, a picture is worth a thousand words, so here's a little graphic that should help clear up what these two effects do:
Here's a little bit more technical of an explanation from the 3DS Max online reference:
Displacement Map: A Displacement map displaces the geometry of surfaces. The effect is similar to using the Displace modifier. Unlike Bump maps, a Displacement map actually changes the geometry of the surface or patch tessellation.
Bump Map: A Bump map uses the intensity of the map image to affect the surface of the material. In this case, the intensity affects the apparent bumpiness of the surface: White areas protrude, and black areas recede. Use a Bump map to take the smoothness off a surface or to create an embossed look...The bumpiness is a simulation created by perturbing face normals before the object is rendered. Because of this, bumps don't appear on the silhouette of bump-mapped objects.
For a tutorial of how to achieve either of these effects using 3DS Max, go to this video tutorial.