The hunt for the Bismarck is such an epic tale, a story of cat and mouse between the British and Germans. There seems to have been a genuine fear on the part of the British regarding the ship, culminating with Churchill issuing the urgent order "Sink the Bismarck!". Yet it doesn't seem as if the Yamato, an intimidating ship in it's own right, was a priority to the US Navy.
Why didn't it inspire the same sort of fear in the American's, and why wasn't there an urgent chase to sink it like there was for the Bismarck?
Well, the simple answer is that the two ships are not parallel, and the goals of the German navy and the Japanese navy were inherently different.
The Bismarck's goal in Operation Rheinübung (Exercise Rhine) was to interdict Allied shipping and supplies to the island of Britain, and the Bismarck was tasked to do this along with the heavy cruiser Prinz Eugen. The operation was a follow up to a similar mission carried out by the German ships Scharnhorst and Gneisenau which highly alarmed the Admiralty, so much so that it repeatedly attacked the ships in harbor at Brest, successfully disabling them. If successful, Operation Rheinübung could have significantly affected supplies of food and material to Britain (and in fact Germany was able to significantly disrupt supply lines mid-war using submarines, in what's termed the Battle of the Atlantic).
In any event, what happened in May of 1941 was that Bismarck and Prinz Eugen sortied from their base at Gotenhafen (now Gydinia) in occupied Poland, intending to break out of the Baltic and raid troop and shipping convoys in the Atlantic.
The ships were sighted near the Skagerrak by a Swedish cruiser, which reported the sighting to the (neutral) Swedish government, whereupon British agents in Sweden sent the information on to the Admiralty. The German ships stopped to refuel at the Grimstadfjord, at which point British forces at Scapa Flow had already sortied to search for the Germans near the Denmark Strait, on the assumption that they would go north around Iceland. The battleship Prince of Wales and the battlecruiser Hood were the first to leave Scapa, followed by the battleship King George V and the new aircraft carrier Victorious.
The German ships were found by the cruisers Norfolk and Suffolk, patrolling near the straits, after which a brief fight ensued that featured Bismarck knocking out its own radar with the concussion of its own guns. (It was not a well-designed ship.). The British cruisers ran out of range and shadowed the Germans with their radars, passing information to the rest of the British fleet, which was converging on the location (even the British Force H, based at Gibraltar, was part of the response). The next action in the sequence of events was the Battle of the Denmark Strait, in which POW and Hood engaged Bismarck and PE, with the result that Bismarck hit Hood near her magazines, and Hood blew up with the loss of all but three hands; POW's gun turrets started to jam and she was forced to break off the action, but not before hitting Bismarck in its forward oil tanks, starting a serious leak that depleted its fuel supplies and also gave the British ships another way to keep shadowing it.
Given Bismarck's leaking fuel tanks, German admiral Lütjens decided to allow PE to go solo into the Atlantic, and attempt to dash back to Brest with Bismarck for repairs. Bismarck was attacked by Swordfish torpedo bombers which hit the ship under the bridge, but caused little damage against the anti-torpedo armor; after this attack, poor weather caused the British to lose track of Bismarck for about a day, until the German battleship broke radio silence to transmit a message to Brest. This allowed the British to triangulate Bismarck's position, and the ship was found again by a flying boat patrolling from Northern Ireland.
At that point (26 May), the British carrier Ark Royal again launched a squadron of Swordfish, which accidentally attacked the British ship Sheffield (part of Force H, which the pilots did not know was in the area). A second strike later that day found Bismarck, and a torpedo hit in her stern disabled the ship's steering.
On the morning of 27 May, the battleships Rodney and KGV attacked the Bismarck with their guns, silencing its fire within half an hour and causing heavy casualties, but failing to sink it (they were probably firing from too close in). The cruiser Dorsetshire made a torpedo attack and scored three hits; German sailors were setting scuttling charges at this time and the Bismarck sank around 10:40 a.m.
Prinz Eugen continued on the raiding mission, but developed engine trouble and was forced to return to Brest by June 1. The overall raiding mission was a failure; the loss of Bismarck represented the loss of 25% of all German capital ships available to them during the war; and the Kriegsmarine never attempted another surface raiding mission during the war.
Yamato and its sister ship Mushashi were the largest battleships ever built, weighing more than and carrying bigger guns than the American Iowa class (and the Iowa's planned successor, the Montana class).
Yamato was launched in August 1940 and commissioned in December 1941, after the Pearl Harbor attacks, and spent most of the war doing nothing in particular -- Yamamoto Isoroku was aboard her during the Battle of Midway, but the ship never came near the actual action, and in fact the only time it fired its guns in anger was during the Battle off Samar, when it was ignominiously chased off by the escort carrier group named "Taffy 3," which consisted of six escort carriers, three destroyers, and four destroyer escorts. (For a sense of proportion, any one of Yamato's turrets weighed more than the DDs and DEs.)
The superbattleships' lack of contribution to the war effort was not unnoticed -- as a freighter officer observed, "We were always being sent to the very front lines, and those battleships never even went into battle. People like us . . . were shipped off to the most forward positions, while those bastards from the Imperial Naval Academy sat around on their asses in the Yamato and Musashi hotels." (the above quoted from Ian W. Toll, The Conquering Tide; original citation "Reiji Masuda, oral history, in Cook and Cook, eds., Japan at War, p. 301.") Yamato was sunk on what was essentially a suicide mission at the end of the war, taking at least 11 torpedoes and 6 bombs.
So having built these monsters, why didn't the Japanese use them? Part of the reason is that they were literally too big to be used much at all -- their fuel consumption was enormous, with each one having 6,300 ton tanks, and the Japanese stocks of fuel oil and refueling infrastructure lagging behind. Part of the reason is that the war in the Pacific was largely an aerial war, fought between carriers and in attacks on islands or on ships using land-based aircraft, where unescorted surface ships were incredibly vulnerable. And part of the reason is that the superbattleships were built to deal a decisive blow in a Mahanian-style fleet action that the US Navy refused to participate in.
If any ships were going to be chased in the Pacific, it would have been the Japanese aircraft carriers -- and the US Navy did exactly that, with the aid of intercepted codes, first at the Battle of the Coral Sea, which disabled Shokaku and Zuikaku; and second at Midway, where Shokaku and Zuikaku's absence contributed to the American victory and sinking of four Japanese carriers (Akagi, Kaga, Soryu, Hiryu.) After Midway, US forces seized the strategic initiative in the Pacific and did not let it go.
There are a lot of reasons one could explore in order to answer this question. Propaganda value of the Bismarck, Geographical limitations of the Yamato and Japanese Battleship fleet, Strategic situation in the Pacific, IJN Fuel Constraints and so on.
To me though, the most interesting reason is the Ascendency of Air Power over Battleships in the period between the outbreak of the war in Europe and Yamato entering service in February 1942. In the beginning of the war, Air and Naval forces had different opinions on how effective air attacks could be against battleships. Initial actions of the war in Europe provided a mix of evidence. Royal Navy and RAF air attacks had been devastatingly successful against Italian battleships at Taranto, but unsuccessful against German ships in waters off Norway. German Battlecruisers Scharnhorst and Gneisenau had success in sinking HMS Glorious, a British aircraft carrier in the North Sea.
However, in 1941, the aviators began running up the score against battleships. Torpedo bombers had crippled Bismarck at sea. Scharnhorst and Gneisenau had been beat up badly at port in Brest by air attacks. The Japanese attack at Pearl Harbor crippled the US Pacific fleet. HMS Repulse and HMS Prince of Wales had been sunk by Japanese Naval Aviation while underway at sea in December of 1941.
Taken together, it had become brutally obvious by early 1942 that Battleships could be effectively countered by naval aviation. Yamato was a large ship that had the potential to be effective in the right circumstances but Japanese naval air was the key factor in the power of the IJN by the time Yamato was operational and naval aviation would continue to increase in relative power for the rest of the war.
