Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$?
Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$?
The list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.
The answer to the first question is no: Golomb [Go70] observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker [Wa76] proved that the equation \[7^3x^2=3^3y^2+1\] has infinitely many solutions, giving infinitely many counterexamples.
See also [364].