Americans are betting on aircraft, the Russians on a new generation of guided missiles
In April 1942 the US Navy carried out a bombing mission of Tokyo using carrier-based aircraft – the famed "Doolittle raid" that was supposed to raise American morale after Pearl Harbor and a succession of other Japanese victories. The parametres of the mission called for the aircraft to take off while the two launching carriers were still 900 kilometres from the Japanese islands - that is to say beyond the striking range of most Japanese aircraft.
Because no standard carrier aircraft had this sort of range, specially modified twin-engine B-25 bombers were selected for the task. These, however, could only take off from a carrier but were far too large to land on one and would therefore have to continue on to China and land there.
Actually the mission was so complicated that in the end 15 of the 16 bombers were lost as they ran out of fuel (the aircraft were launched 300 kilometres earlier than planned) before reaching their bases in China and their crews parachuted over Japanese-occupied territory.
Clearly the plan was asking a lot of the air crews as the the mission planners prioritized keeping the risks for the valuable aircraft carriers at a tolerable level. Sneaking up on the Japanese home islands was risky enough, but entering into the range of Japanese land-based bombers would have been downright mad.
The aircraft carrier was a formidable weapon that in WWII completely outclassed the battleship and consigned it to history. Its aircraft could discover and attack a battleship long before the battleship could discover the carrier or get it within the range of its guns (usually around 30 kilometres). At the same time it was understood a carrier stood no chance against a concentration of land-based aircraft and needed to at all times stay beyond their range.
This situation lasted into the late 1960s when the Soviets realized that advances in missile technology may have made "surface combatant" vessels competitive again. In the following decade the Soviet union started laying down large surface ships built around their new powerful long range anti-ship missiles. These "rocket cruisers" needed to come no closer than some 600 kilometres to deliver their payload and thus could in theory go against a carrier and hope to come out on top.
Specifically, the Soviets built the smaller and cheaper Slava-class cruisers and the much larger nuclear-powered Kirov-class "battlecruisers". All three Slava-class ships which were completed in the Soviet era continue to be operated by the Russian navy today. The fate was not as kind to the Kirov-class vessels which are much more costly to operate. Three out of the four built were laid up during the cash-strapped 1990s.
Advances in missile technology coupled with an improving fiscal situation, however, again came to benefit the Kirov class. Thinking in the late 2000s how to best improve their naval capabilities the Russians couldn't think of anything better but to bring back the unused Kirovs as platforms for a new generation of anti-ship missiles.
Russians initially explored the option of bringing back all three mothballed Kirovs but in the end settled for retrofitting and reactivating only the newest of the them, the Admiral Nakhimov (formerly Kalinin). The rearming of Nakhimov is scheduled to be completed in 2018 after which the currently active Kirov-class vessel Pyotr Velikiy (ex-Yury Andropov) will go in for the same procedure and re-enter active service in 2022.
The key thing to take away is that when the two vessels rejoin the navy in 2018 and 2022 they will be a far different beast from the (already formidable) Kirov the Russian navy has now. Their anti-aircraft defenses will be updated from the S-300 to the S-500 standard and their main armaments completely overhauled. Their arsenal of the 1970s P-700 missiles which have a range of some 600 kilometres and travel at Mach 2.5 will be replaced by a mix of (subsonic) Kalibr cruise missiles whose range is measured in thousands of kilometres and 3M22 Zircon anti-ship missiles which travel at speeds of Mach 5 or 6.
The Kirov-class vessels which weigh in at 28,000 tons and are over 250 metres long suffer from some of the same vulnerabilities that carriers themselves do – they can be sunk by aircraft or submarine. This is the reason the Soviets despite deeming them "cruisers" never intended for their large missile carrying vessels to operate alone, or to conduct romantic, but suicidal, solo raids against carriers. Instead they were envisaged as centerpieces of a larger surface group comparable to that accompanying the carrier itself.
A Soviet Kirov-class led surface group could in theory hope to successfully attack and wreak havoc on a carrier group (especially seeing how Cold War military planners seriously considered using nuclear-tipped missiles), but its real value for Russia in a stand off against a more powerful navy would likely be as a deterrent and area denial asset.
A Kirov moving at the far end of the range of friendly land-based anti-air systems and land-based strike aircraft would be reasonably protected from both carrier aircraft and submarine attack, but would at the same time greatly extend the reach of what would otherwise be land-based anti-ship weapons.
Now, it is the case that the Russians have managed to fit the Kalibr cruise missile on ships as small as Gepard and Buyan-class corvettes. This means even their smaller vessels provide some area denial. However, these remain bulky weapons so that smaller vessels like frigates and destroyers come with only a few launch pads and rounds at a time.
However, the Gorshkov-class only carries 2 launchpads for anti-ship missiles and cruise missiles, and Steregushchy-class has none. A Kirov-class carries 10 such launchpads (each with eight rounds) in addition to assortment of other weapons. Altogether it is a floating platform for over 300 missiles.
Basically this means a Kirov class missile ship, covered by friendly aircraft, can shut down an entire sea against enemy fleet encroachment – something that could never be accomplished by 2.5 frigates. Moreover against a weaker or equivalent navy the ocean-going cruiser can take the fight to the enemy but a frigatte can not.
Since these ships have already been built it is now cost effective to use them rather than build new ones.
The battlecruisers were indeed outdated even sooner than the standard battleships (and were really conceived to get around treaty limitations rather than because they were a sensible design). But that's missing the point that a sleek Russian missile carrier concept has very little do to do with a WWII gunboat.
A vessel like the Russian heavy missile ships has never been tested in war and there's reason to believe it is at the very least no more outdated than the aircraft carrier which is still many times more expensive.