Differences between gel and gelatine

A gel is any liquid (usually) or gaseous medium suspended in a solid three-dimensional mesh which entraps the medium so that it does not flow.

By way of (somewhat flawed analogy) think of a giant role of bubble wrap. Its mostly air. But the plastic keeps the air from flowing at a large scale.

Gels can range from very soft to very hard. New modern aero-gels have fascinating properties.

Gelatin is a protein formed when collagen (a connective protein in animals) is heated under moist conditions. Gelatin forms cross-linked networks exceedingly well, and therefore is a wonderful gelling agent.

The name has also taken to mean the gel created from gelatin and a water-based liquid like stock or sugary flavored water. You may have guessed by now that gels in general take their name from gelatin.

So the specific answer to the first part of your question is: gels are the general category, and gelatin (in the second sense) is a specific kind of gel.

I have never heard the term gellification. The only verb I am familiar with is to gel. The process is gelatinization or (more familiar to me), just gelling.

In those links in Wikipedia for gel, gelatin, collagen, and areogel you will find out much more.