Step 1: Picture what an orbital actually is.
An orbital is a region of space described completely by a fixed set of the first three quantum numbers. So before we even talk about the electrons inside it, the orbital itself locks in three values.
Step 2: List the three numbers fixed by the orbital.
The principal quantum number $n$ fixes the shell (size), the azimuthal quantum number $l$ fixes the subshell shape, and the magnetic quantum number $m_l$ fixes the orientation. Once we say two electrons sit in the same orbital, all three of these are automatically identical for them.
Step 3: Recall the rule that governs electrons.
The Pauli Exclusion Principle states that no two electrons in one atom can carry the exact same set of all four quantum numbers $n, l, m_l, m_s$.
Step 4: Find where the difference must hide.
Since $n$, $l$ and $m_l$ are already forced to be the same for both electrons, the only number left that can differ is the fourth one, the spin quantum number $m_s$.
Step 5: Confirm the spin values are valid.
The spin quantum number takes only two values, $m_s = +\tfrac{1}{2}$ and $m_s = -\tfrac{1}{2}$, which neatly allows exactly two electrons per orbital and no more.
Step 6: Match to the options.
Principal, azimuthal and magnetic quantum numbers are identical, so they cannot distinguish the pair. The distinguishing factor is the spin quantum number, option (4).
\[ \boxed{\text{Spin quantum number distinguishes the two electrons}} \]