Long story short: Things can only look as good as the math used. So deficiencies in mathematical models are one of the #1 reasons.
There are many reasons. Motion: Modelling physics of motion on a 3D model is very difficult to do and still have it look 'natural' because a 3D models and the details are 'rigid' set of fixed points wrapped in textures. It's hugely a matter of processing power, rendering technology and time and money spent on the art.
CG characters look CG as a function of the state of rendering technology, characters are not composed of discrete atomic units in 3D modelling software, they are essentially fixed point flat 3D objects (triangles/quads/etc) wrapped in textures. So you can't get the subtlety of human motion yet without finding a good algorithms to model it correctly, also due to the rigid body nature of 3D characters there is complex physics that happens when you move your arm that just doesn't effect your arm per se, but also your skin, the muscle beneath your skin, the pumping of your blood through your veins, etc, that all effect the motion of many parts of your bodies skin and sometimes other muscles. We're years away from rendering the types of complex realism you'd see in a real world "iron man" competitions where you see these ripped people with bulging muscles and veins.
The resolution problem: The smallest "detail units" and maximum resolution in a 3D package can't compete with the real world atoms (i.e. think distance detail and clipping problems). If you look at skin of the Final fantasy characters you notice the skin textures and the models aren't detailed enough. If you really pay attention when they are in close the skin on both aki and the guys are totally flat and uniform!, real peoples skin is not flat, its bumpy and pourous and dirty and wrinkly/stretchy, try looking at your skin in the light from different angles having the light reflect off it then push and pull and stretch your skin, then go back and look at CG characters and pay attention if you see any "realistic" stretching. Real peoples skin and arm hair don't look "flat" or "matte" either, if you not notice all the characters in final fantasy have 'the same' uniform look. Real peoples skin asbsorbs/reflects light differently. i.e. hold up a flashlight to your fingers and you see that light passes through your flesh so flesh is transparent and lets a certain degree of light pass through it.
The lighting models and color alogirthms they use, make the skin look not real. They can't seem to compensate for proper 'material' light absorption/etc. As you already know modeling clothes and folding skin is difficult, part of the reason the characters look so plastic and don't animate "right" is because their 'clothes' and 'skin' are super rigid and 'hard body', but this largely depends on how the characters in each scene were modelled, what artists worked on them, and a host of other factors. Some characters clothes are part of the model itself and not an actual seperate object (those "cheating", time and money saving, artists/people don't help!).
It's not an inherent problem of computer drawings per se, its the amount of details big and small your rendering technology can reproduce when compared to the real world. Things in the rendering world have to be mathematically modelled, so things can only look as good as your math. The resolution of the pictures of final fantasy are very high but your eyes resolution and that of the real world is much much higher, approaching infinite detail. So even if CG characters even get to start looking real, they will still look "off" simply due to the fact that the world approaches levels of detail that cann't be accurately captured or modelled yet by current rendering tech.