"The special relationship" the term used by Winston Churchill to describe the uniquely close connection between the United States and the United Kingdom. The relationship between the United States of America and the United Kingdom goes back almost two hundred years before the United States declared independence from Great Britain. In order to understand American and the British political system, firstly the US constitution, the type of political parties and the presidency system will be conceded. Secondly the British constitution, the type of political parties and the prime minister system will be also considered. Lastly the differences between the political scenes on the two sides of the Atlantic will be also looking at.
As define by Adrian Leftwich politics are at the hearth of all collective social activities, formal and informal, public and private, all human groups, institutions and societies.  This means that politics takes place at every level of social interaction.
US constitution is the oldest written constitution and holds a central place in United State law and political culture. The Constitution in 1787 forms the foundations of the United States federal government and created a structure.  This has remained an active governing force through the changes in US society and a very remarkably stable document "the American constitution is the most wonderful work ever struck off at a given time by the brain and purpose of man"  . The Constitution creates the three branches of the national government: a legislature, the bicameral Congress; an executive branch led by the President; and a judicial branch headed by the Supreme Court.  The Constitution specifies the powers and duties of each branch.
The good thing about the US constitution is the underlying acceptance that it could be wrong. If you don't like the laws, it's OK to rewrite them, provided you have enough support and the balance of power between the Judicial/Legislative/Executive.
US political system is dominated by two political parties, which are the Democratic Party currently positions itself left-of-centre in American politics and supports an American liberalism platform and the Republican Party, which positions itself as a right-of-centre and supports an American conservatism platform.  The reason beyond these two parties is that the electoral system is 'first past the post' or simple majority which, combined with the large voter size of the constituencies in the House and the Senate, ensures that effectively only two parties can play. The other reason is the influence of money. Money has tremendous influence on the political process and campaign, there more you have their more you can reach out that is why the US can only 'afford' two parties or, to put it another way, candidates of any other party face a formidable financial barrier to entry.
US presidency system is often referred to by the media as the White House. The US President is both the head of state and the head of government; He presides over the executive branch of the federal government, and is the commander-in-chief of the United States armed forces. Within the executive branch, the President has broad constitutional powers to manage national affairs and the workings of the federal government and may issue executive orders to affect internal policies. The total electoral vote is 538 so to become a president in the US you need 270 electoral votes  . The President is elected for a fixed term of four years and may serve a maximum of two terms. The President is not elected directly by the voters but by an Electoral College representing each state on the basis of a combination of the number of members in the Senate two for each state and the number of members in the House of Representatives.
Unlike the US the UK has no core constitutional document or unwritten constitution there is no single document that can be classed as Britain's constitution. The UK constitution however has traditionally been the doctrine of Parliamentary sovereignty, according to which the statutes passed by Parliament are the UK's supreme and final source of law, the Parliament can change the constitution simply by passing new Acts of Parliament. 
UK political parties are an important feature of the British political system. Tree groups dominate party politics in Britain, the conservative, the labour parties and the Liberal Democrat. In addition to these three main parties, there are some much smaller UK parties, most of these parties are transitory but a few have endured sufficient time to merit examination. Some are significant such as the Nationalists in Scotland and Wales, their support has been considerable in particular areas of the United Kingdom.  Each party have their own institution, it chooses its leader but all involve all the Members of Parliament of the party.  The leader of the political party with the largest number of members in the House of Commons becomes the Prime Minster.
UK does not have a President. Constitutionally the head of state is the monarch who is a hereditary member of the Royal Family. A historic feature of the UK constitution, the Royal Prerogative gives the Crown (the monarch) special powers, including the power to declare war, to make treaties, to pardon criminals, and to dissolve Parliament. Today the role of the monarch in such matters is largely ceremonial, but the Royal Prerogative gives considerable powers to government ministers acting on the Queen's behalf.  So the most important person in the British political system is the Prime Minister. In theory, the Prime Minister simply chooses the ministers who run Government departments and chairs the Cabinet - the collection of the most senior of those Ministers. In practice, however, the Prime Minister is a very powerful figure and increasingly has been behaving much like a president in other political systems, especially in the area of foreign policy.
The most fundamental difference of UK and US political system is the constitution - or the lack of one. The US has a written constitution which is very difficult to change and UK does not have a single document called the constitution but instead its constitutional provisions are scattered over various Acts of Parliament, any of which can be changed by a simple majority in the Parliament. Like Thomas Paine once said "Government without a constitution is power without right" this means the British government can make the constitution as whatever the wants it to be.
There are three major political parties in the British political system: The Labour Party the centre-Left party, The Conservative Party the centre-Right party and the Liberal Democrat the centrist. Where as the US political system is dominated by two political parties: the Democratic Party and the Republican Party. The division between the Labour and Conservative in the UK is the same as Democratic Party and the Republican Party in the US. Each country, one political party is characterised as Centre-Left and the other as Centre-Right.
The US President holds the position of Head of State as well as Head of Government. The UK Prime Minister, it would appear, has more influence in domestic, able to dominate his part, legislature and executive branch. The US President, on the other hand, appears to have the position of supremacy in domestic politics, He does not hold the same position of power in domestic affairs as the UK Prime Minister, but his position of strength appears to be in the realm of foreign and international matters.
In conclusion by referring to the question whether British and American politics are different, we can see that the term "special relationship" as nothing to do with the way these two countries is lead. In each four areas we see how both countries have diverse ways of viewing politics.
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