I respectfully submit that the above post is incorrect. Although the Black Sea does give Russia access to the Mediterranean, it is not its only access to the sea. Russia has had access to the Baltic Sea through the Gulf of Finland and has had that access since its defeat of Sweden in the Great Northern War of 1700-1721. The important issue was not access but control of accessthrough the Strait of Bosporus and the Dardanelles.
Although the question does not specifically mention the Crimean War, that war was fought primarily over control of the Crimea which would allow Russia a commanding position in the Black Sea and allow it to capture Constantinople itself, one of the goals of Czar Alexander II who hoped to return that city to Eastern Christianity from the Turks. This would upset the delicate balance of power which existed in Europe; and as a result, England and France went to war with Russia on the side of Turkey to prevent Russian control. At present, both Russia and Turkey have access to the Black Sea; however entrance into that Ocean by means of the Bosporus is under Turkish control. Were this Russia's only access to the world's oceans, it would be at the mercy of Turkey, a situation which it would never allow.
For more information on the diplomatic situation which led to the Crimean War, I recomment Orlando Figes' highly readable book, The Crimean War: A History.
The Black Sea has always been more important for Russia than for Turkey. The reason for this is partly economic, but is also military. In both cases, the importance was due to the fact that Russia needed the Black Sea in order to gain access to world shipping lanes.
Without access to the Mediterranean through the Black Sea, Russia can be cut off from the Atlantic Ocean. Russia has no ports that are ice free all year. This is a major disadvantage both in economic and in military terms. The Black Sea is not nearly so important for Turkey because Turkey borders directly on the Mediterranean. By contrast, Russian shipping has to go out of the Black Sea and through the Dardanelles (very narrow, controlled by Turkey, easy to shut off) to gain access to the Mediterranean.
For these reasons, Russia has always wanted to control the Black Sea and, more importantly, the outlet from the Black Sea to the Mediterranean.