$P/poly = NP/poly$ implies $NP \subseteq P/poly$, which in turn has interesting consequences like the collapse of the polynomial hierarchy.
Are there interesting implications for $P/poly \neq NP/poly$?
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Emil Jeřábek' comment answers the question:
P/poly $=$ NP/poly is equivalent to NP $\subseteq$ P/poly
Note the corollary
P/poly $\neq$ NP/poly implies P $\neq$ NP.
Proof of corollary:
Proof of Emil's comment: It is sufficient to show that NP $\subseteq$ P/poly implies P/poly $=$ NP/poly.
All the above proofs relativize, because the existence of NP-complete problems is also true in relativized worlds. This suggests that it is fruitless to search for a proof that P/poly $\neq$ NP/poly. However let's summarize the removed motivation section from the question as "The advice string could be a formal axiomatic system (automatically guaranteed to be consistent, evil grin) whose strength is quickly increasing with input length, and NP is extermely good at exploiting this advice." If one is not very careful that "existence of a sequence of advice stings" only has "formal" meaning relative to a fixed formal system, that setup is likely to allow the construction of apparent paradoxes. But the construction of such paradoxes might be fun nevertheless, and perhaps they might even suggests ways how to construct independence proofs (for sufficiently weak formal systems).