Pivot

What is the differences between pivot in manipulation window and pivot in visual tools?

Visual Tools pivot moves the pivot to the center of the object bounding box or mesh.

Manipulation is for fine tuning if you want the pivot to align or be snapped to a specific point in space.

Basics:-

With an object selected, enable "Visual Tools > Pivot". The object pivot is seen as the manipulator, and you can move the object pivot by using the manipulator, be it by dragging or snap points. The options also allow setting the pivot to object center. There is more advanced tools for setting object pivot:- "Streamline Tools > Object Pivot" and "Streamline Tools > Object/Mesh Pivot"
Each object as its own pivot location/orientation.

The pivot in manipulator window, is the manipulator. It can be set using the "Set" buttons in the manipulator window, or there is a more advanced manipulator settings tool:- Streamline Tools > Set Manipulator"
The position/orientation of the manipulator changes depending on what setting are made in the Manipulator window (selection/object/manual etc).

To add to possible confusion, there is also a group Pivot, that can be set via "Streamline Tools > Group Pivot"