The difference between Ampere's law & Biot-Savart Law

[PhiowPhi]

When considering the magnitude of the magnetic field at a certain point ($P$) away from an infinite/finite wire, I can't understand how an infinite wire would generate a stronger magnetic field ($B$) in contrast to a finite wire that has the same dimensions and current applied, at the same point ($P$). I understand how to use the equations, yet I can't fully understand the concept.

Considering the case of an infinite wire:

At the point the magnetic field is point out of the page.
Case of the finite wire:

Where (A) is the current element region of focus.
Why are the different considering the same $I$ and dimensions for the two?
I found from other sources that all the current elements in the wire would contribute to the magnitude at point($P$) but how is that so? If the magnetic field created by the current element loops around it:

If so, how can regions B,C contribute to point($P$) for the two wires?

For the finite wire:

 

[mfb]

If the magnetic field created by the current element loops around it:

That is not the full magnetic field induced by the wire element. The full magnetic field is 3-dimensional, it just has its strongest region orthogonal to the wire.

You can use Biot-Savart to calculate that.

 

[PhiowPhi]

The full magnetic field is 3-dimensional, it just has its strongest region orthogonal to the wire.

For some reason I can't imagine that...