The Iowa class is almost twice the tonnage of Kirov-class.
If we were to compare the munitions on these ships in late Cold-War era, the Kirov has an advantage on-paper with its long-range, supersonic anti-ship missiles. The Kirov also has superior missile defense systems. The Iowa has better armor but I doubt it'd have survived a barrage of P-700 missiles without being mission-killed.
I doubt the original Iowa-class BB's would've been suitable (or cost effective) for a complete conversion to modern "guided missile BB". But if both the US and Russia were to construct brand new ships today and equip them with their best hardware, the US would have an edge in sensors and AAW systems.