Such difference between Edge Split as modelling tool and modifier Why Edge Split doesn't dynamically apply its functions through animation?

Using it as a tool (Mesh->Edges->Edge Split) allows you to manipulate each face independently. However Edge Split modifier doesn't work like that. Yes it still increase vertex count but doesn't actually split geometry, unless click Apply button.

Setup:

1. Hook is binded to the top face with Sharp edge surrounding

2. Split Edge modifier is splitting top face



Now animated hook is trying to move it: There is no visible split.


After pressing Apply in Edge Split modifer: Whole different picture
so why it splits edges and at the same time keeps them connected
  before applying?


Edge split modifier actually split geometry. But you test this using wrong way. Try to add Susurf after Edge Split:



When you adding hooks it moves both arrays of vertex, that splitted by edge split modifier