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Galactic nuclei are promising sites for stellar origin black hole (BH)mergers, as part of merger hierarchies in deep potential wells. We show thatbinary black hole (BBH) merger rates in active galactic nuclei (AGN) shouldalways exceed merger rates in quiescent galactic nuclei (nuclear star clusters,NSCs) around supermassive BHs (SMBHs) without accretion disks. This isprimarily due to average binary lifetimes in AGN that are significantly shorterthan in NSCs. The lifetime difference comes from rapid hardening of BBHs inAGN, such that their semi-major axes are smaller than the hard-soft boundary oftheir parent NSC; this contrasts with the large average lifetime to merger forBBHs in NSCs around SMBHs, due to binary ionization mechanisms. Secondarily,merger rates in AGNs are enhanced by gas-driven binary formation mechanisms.Formation of new BHs in AGN disks are a minor contributor to the ratedifferences. With the gravitational wave detection of several BBHs with atleast one progenitor in the upper mass gap, and signatures of dynamicalformation channels in the $\chi_{\rm eff}$ distribution, we argue that AGNcould contribute $\sim 25\%-80\%$ of the LIGO-Virgo measured rate of $\sim 24\rm{Gpc}^{-3} \rm{yr}^{-1}$.

Binary Black Hole Merger Rates in AGN Disks versus Nuclear Star Clusters: Loud beats Quiet