This is partly motivated by the question of whether or not Jerusalem is Israel's capital, but my question is broader.
Why should a state have a single, well-defined "capital city"? Why does it matter whether of not other states "recognize" this city as the state's capital? What does it mean to be a "capital" of a state?
Of course, there are the practical considerations: the "capital" is where the government institutions are located – the residence of the head of state, the legislature, the main ministries, the supreme court etc. But that is a very "factual" definition — it does not depend on any formal recognition. By this definition, (West) Jerusalem would be the actual capital of Israel, since the main government institutions are situated there.
But why, then, does it matter which city is formally declared the capital of a state? And why does it matter whether this is formally "recognized" by foreign states? There are quite a few anomalies: in the Netherlands, Amsterdam is the official capital of the state, yet most government institutions are located in The Hague. (All foreign state recognize this; yet would it matter much to the Dutch if a foreign power stopped recognizing Amsterdam as the capital, and declared that The Hague is the "true capital"?) On the other hand, South Africa appears to have several capitals: the administrative capital is Pretoria, the legislative capital is Cape Town, and the judicial capital is Bloemfontein. And in Switzerland no city has the official status of Capital City. So why does it matter that a state should have a single formal "Capital"?