Why is quinone mono oxime more stable than 4-nitrosophenol?
Two factors come into play: (1) the inherent difference in resonance energy is not as great as you might think, and (2) the greater tendency for a dissociable proton to bind more tightly to an oxime function rather than a phenol function appears to counterbalance the reduced $\pi$-electron bonding difference.
Less than meets the eye
Suppose you were to use the Hückel model to estimate the pi-electron binding energy for several eight-carbon conjugated systems.  By that method I find the following results:

For the conjugated straight chain $\ce{CH2=(CH=)_6CH2}$ the total $\pi$-electron binding energy works out to $9.52\beta$, where in this theory $2\beta$ represents one isolated $\pi$ bond.

For styrene, with a $\pi$-electron pattern matching the nitroso-phenol, the result is $10.53\beta$.  A difference of about one $\beta$ unit is typical when an aromatic ring is compared with the opened ring.

For para-xylylene whose quinonoid $\pi$-electron structure matches the oxime tautomer described here, we get $9.92\beta$.


Of course the aromatic structure is most stable, but not all non-aromatic structures are created equal.  The Hückel-model calculations indicate that even though the quinonoid structure is typically called non-aromatic, in fact it captures a significant part (about $40$% based on the numbers) of the extra $\pi$-electron stabilization we usually associate with an aromatic ring.  The energetic difference between the two competing $\pi$-electron configurations is thereby less than the &quot;aromatic&quot; and &quot;non-aromatic&quot; labels would seem to imply.
Swimming upstream
Now let's see where the dissociable proton is more stable.  To do that we look up the $pK_a$ values typically associated with each prospective acidic functiin:

Phenol gives $pK_a=9.95$.

Acetoxime, $\ce{(CH3)_2C=NOH}$, gives $pK_a=12.2$.


The oxime function is clearly holding the proton more tightly; placing the acidic proton there instead of on the phenol function saves $2.2$ powers of ten in the equilibrium constant.
We cannot directly compare this with the Hückel-model calculations from the previous section, because that model is only semi quantitative and a more complex calculation would be needed to capture the effects of the heteroatoms in the conjugated structure.  But the results do suggest that the combination of the quinonoid structure capturing part of what is usually aromatic $\pi$-electron stabilization with the preference of a dissociable proton to bind to the oxime function leads the oxime tautomer to win out over the nitroso-phenol.