A fermion is described by a set of quantum numbers, this set of numbers lead us to a unique wave function. If two fermions are described by the same wave function (violating the Pauli's exclusion principle), how can we differentiate this fermions if experimentally it produces the same result?
[Physics] Experimental evidence of Pauli’s exclusion principle
experimental-physicspauli-exclusion-principlequantum mechanics
Related Solutions
I'll try to give a qualitative view. There are an array of forces working together at various distances and strengths that stabilize bulk matter. The Pauli Principle could probably be considered to be one of the lowest fundamental levels.
The Pauli Exclusion Principle is often confused with the cause of macroscopic effects like being responsible for atoms or molecules not occupying the same space, but that is not really the full picture. Atoms and molecules after all are mostly empty space. The exclusion principle is only partly responsible for why macroscopic scale matter can't be in the same place at the same time.
And the stability of electrons themselves in an atom are unrelated to the Pauli Exclusion Principle which is strictly about quantum states of fermion matter. In this respect fermion matter must occupy some finite volume. The electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells cannot be squeezed too closely together.
Andrew Lenard considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle. But this doesn't mean you can't compress bulk matter with millions of atoms and molecules tighter together you just have to overcome the other repellent forces first. While the Pauli Principle sets the ultimate limits on all the bits that are fermions.
The spin wave function is symmetric with respect to the exchange of particles. Therefore the spacial wave function has to be antisymmetric. I.e. at least one of the quantum numbers has to be different.
The wave function may look as if the electrons have opposite spin, but actually the spins are the same if measured at an axis 90° from z.
EDIT:
The spin eigenvectors of different axes are not independent from each other. $$ \begin{align} \left|\strut\uparrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\\ \left|\strut\downarrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) \end{align} $$ Substitute that into your definition of $\left|\strut\psi\right\rangle$ and you will get $$ \begin{align} \left|\strut\psi\right\rangle &= \frac{1}{2\sqrt 2}\left[\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) + \left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ \end{align} $$
To check for the symmetry you don't need this calculation. It is sufficient to check that $$ \frac{1}{\sqrt2}\left( \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle + \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle \right) = \frac{1}{\sqrt2}\left( \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle + \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle \right) $$
The Pauli principle says that wave functions have to be negated when swapping any two Fermions.
Best Answer
I think the atomic configuration is the strongest evidence to Pauli's Exclusion Principle. You simply have atoms that have energy levels sorted in a way that agrees with the pauli's principle.
Besides, the Zeeman effect shows the separation between different spins, which raises the degeneracy between the levels that agree in all quantum numbers but not the spin.