I How Do Magnetic Forces Differ Between Magnets and Wires?

[nemuritai]

I understand the iron fillings become little magnets all pointing in the same north south direction similar to the spin aligned electrons in the permanent magnet.
Similarly, a compass near a wire traces out the magnetic field lines ie North/South.

My question is how do I reconcile the fact that the force of a magnet on another magnet is along the field lines whereas for two wires it is perpendicular? The latter is from the fact that the lorentz force is perpendicular to v and B, but why is the force of two magnets alongside the magnetic field instead of perpendicular?

Is the force alongside the B field in one and perpendicular in the other? What error have I made?

 

[kuruman]

An iron filing has a magnetic moment that experiences a torque that rotates it so that its long axis lines up with the local field lines. The force on it is along the direction of the magnetic field gradient which is not necessarily along the local field lines. In the case of the bar magnets that you show in the photo, the magnetic field and its gradient are roughly in the same direction. In the case of the infinite wire the magnetic field is in the azimuthal direction but its gradient is in the radial direction.