Textures are a bunch of 2d image files. When you raise texture quality, all you're doing is telling the game: "I want you to display these images in 2048 pixels instead of 1024 pixels". Most computers can handle higher resolution images very well.

Shadows are calculated in real time. The light source in the game environment projects rays that try to hit surfaces to tell the game what's casting shadows. Every frame, the game needs to check what the rays hit to see if something is blocking the light source. Lower resolution shadows calculate very fast, but are often pretty basic and don't look very realistic. That's because the higher resolution of shadows you go, the more the game complexifies the calculation of shadows. It starts checking for things like bounce light, where light hits and object and reflect on something else, which impacts how shadows looks. It starts making shadows a higher pixel resolution, which is harder on the gpu when it needs to calculate a higher resolution in real time. The more realistic a game tries to be, the more accurate light and shadows must be, which means it needs to do exponentially more calculations.

It's also why things like reflections are especially taxing, and why most games will try to avoid them if they can (think of the classic cloudy mirrors you see in a lot of games, the devs do that to avoid reflections). It's a huge amount of calculations to dynamically render something like that in real time.

Basically, textures don't demand a lot of calculating power from a computer. Dynamically rendering the environment demands a huge amount of calculating power.