During the war, the US eventually became an ally against a common foe, the Germans. In the 1950s, the Soviet Union became our common enemy, and, trying to rekindle the wartime success, Winston Churchill coined the phrase while speaking in the US. Ever since, whenever we have a common interest, or same goal, the phrase 'Special Relationship' is trotted out.
Reagan an Thatcher famously revived it, (but it didn't stop him invading a commonwealth country without telling her). Bush Snr and Major over the Gulf War. Blair and Bush W over Afghanistan and Iraq.
It exists in as much that we do share common objectives with the US. We share a vaguely similar language and culture, so we express our likes and dislikes in a similar way.
The desire to change the regime in Syria has long been the wish list of the British. Obama was fearful of repeating past mistakes of his predecessor and so has stayed out of Syria too long. He has dithered for 2 years, and in the void, all kinds of nasty elements have joined the rebel side. It has turned into a monster.
So we now find something to dislike on both sides. It is unclear why Obama has changed his mind (if he has) but he appears to now want to help the rebels even though they now include, al qaeda, mujahadeen, the muslim brotherhood, Jhiadists and other undesirables.
So in summary, the 'Special Relationship' is just a convenient tag line when both countries have a common purpose. A different tag line is used on different peoples when forming a coalition. 'Our oldest ally' is France. 'Our trusted ally in the east' for Turkey etc.