Why would super resolution using deep learning beat the old school techniques?
The objective is to derive a higher resolution image from a low resolution image. When performing this upsampling, inevitably, you have to fill in information that you do not have.

The differences between the two approaches are in how you fill that information in.

In the standard techniques an underlying model is assumed. So, for example, in nearest neighbour interpolation we decide to fill in the unknown pixels with the value of the nearest pixel. An improvement over that is linear interpolation where we derive a weighted sum of the pixel values around the unknown pixel to set its value. An improvement over that are spline and cubic interpolation which are trying to derive the value of the unknown pixel from a bigger patch of known pixels around it.

In all of these cases, we are assuming a model for filling in the pixel values which is based on physics (e.g. due to the Point Spread Function) or assumptions.

But, how about deriving the "model" that constraints the values of the pixels from the images themselves and then use that model and the limited known data to guess the unknown pixels? Instead of deciding on an interpolant arbitrarily, we derive it.

This is an optimisation problem where we are trying to discover the function that connects the low and high resolution image versions. To do this, you obviously need both the high and low resolution images. But what you can do, is take (known) high resolution images, degrade them (reduce their resolution) and workout the relationship between the high resolution and low resolution. 

The surprising result here is that images depicting different content end up having very similar interpolants. Of course, content does play a role in the sense that if you derive models from a set of images with a specific bandwidth, you will discover models that support that specific bandwidth. You cannot train the model on smoothly varying images with lots of low frequency content and expect it to reconstruct high contrast high details parts of unknown images.

Therefore, the technique is more suitable to classes of images and so far it has been applied to medical imaging (and of course Magnetic Resonance Imaging where there is a long history of reconstruction from limited data) and radio astronomy

For more information you might want to see this and this.

Hope this helps.