Yamato would win, but just.
I know all of you know Yamato's armor, Iowa's armor, guns, ect. And I am not going to add anything in that respect. My meager inention is to give a few details on what people are saying.
First, Iowa's seeming godly radar fire control for her 16 inch guns. It is a huge aid in ship engagements, but it is not a fool proof, I have and I win, tool. Iowa used the same fire control system as the South Dakota, and (though i forget where) South Dakota was struck by a 5 inch shell, and the fire control was disabled. So, in an engagement with Yamato, Iowa would (eventually) get hit by Yamato and the HE shell would utter disable the FCS. Forcing Iowa to either run away, or close range to use their woefully painful optical range finders, which the Japanese outclassed.
Second, the myth that Yamato could not a barn from the inside all due to one incident when Yamamoto was doing a fleet inspection. It is true and well documented, but when I was 9 I shot my first rifle, I was terrible, but after the past 11 years of use and training, I got better! Wow! What a concept! So to say Yamato could not hit anything due to a post launch mishap is unfair. And just for your consideration, Gambler Bay was struck by something that made a much larger hole than any 14 inch or lower could make, and the closest Yamato ever came to GB was about 30 000 yards, so...... Yamato could hit Iowa, maybe after a few rounds but, hey.
Third, that Yamato's guns were actually less powerfull than the iowa's. This is simply not true, Yamato's recorded Shell velocity was close to Iowa's and about 20% heavier (Yamatos shells varied in actual wheight a bit) Also, Yamato did have excellent AP shells. (not as good quality as Iowa, but very close) Also, every one says that based on an experiment done by the US on a turret to see if Iowa could penetrate Yamato's armor is full of holes, I read in a book that the author put alot of time at working the degrees of the armor (a turret) vs the degree of the actuall impact vs how high Iowa's guns could aim. He pointed out that for the turret armor peice tested, the Yamato would have to have had a 30 degree list towards Iowa and Iowa have a 5 degree list away from Yamato. But, I am dubious if Yamato could survive a list of that caliber.
So, here is how I see the battle playing out. Yamato and Iowa cross paths alone. (some how) Iowa detects Yamato and opens fire. First volley is folley as the FC is not as great as modern systems. This obviously alerts the Yamato who sends up a recon plane and aims at the muzzle flashes on the horizon of the next volley which a few shells hit Yamato and do minor deck damage. Using the Spotter plan and he own rader range finders, Yamato manages to score a hit (by the 5th volley) Which takes out the Iowa's FC system. Now is pivitol, does Iowa close range to use her Optics, or run away? Well, iowa turns tail and runs, yamato has a 10% chance of killing Iowa with a few shells to the stern profile while Iowa can not harm Yamato at this range. Congrats Iowa fled. But, if iowa closed range, her fate would be sealed. Iowa would still have to get close enough for her Optics, letting Yamato get in a few volleys, easily killing Iowa here, and as Iowa closes the range, the more favor I have to Yamato. This play out, I give Yamato 65% chance for victory, Iowa 25% and the rest for sinking each other or fleeing.
They are both very good battle ships, and both demand respect, but, there is an old saying in the Automotive world and Battleship world: "There is no replacement for displacment."
P.S Some one mentioned Yamato shooting her guns at planes. This was not foolish, the Japanese had developed Flak guns for the 18.1 inch, the developement stage was really focused on making a shell explode mid air in semi equal parts (like all Flak). Yamato could not "train" her guns to follow planes (the turrets moved too slow) but you aim the guns to let planes fly in a kill zone and than let em have it. The Japanese proved it usefull, but waiting for the planes to go where you want them is not a good way to go I think.