I. 1689 UK and 1787 US : new political regimes (docs 1, 3 & 4)

1. Political regimes

  In the UK In the US
Head of state = A hereditary monarch who inherited the crown. = A president elected by the people, the population.
=> Political regime A monarchy = a political regime in which the head of state is a hereditary monarch. A republic = a political regime where the head of state is a president elected by the people.

2. The separation of powers

  In the UK In the US
The executive power
implements the law
held by the monarch
in Buckingham Palace
+ Chief Minister
in 10 Downing Street
+ the Cabinet
held by the president
in the White House
+ the vice-president
+ the government.
There is no PM in the USA
The legislative power makes law held by Parliament
in Westminster Palace
held by Congress
in the Capitol
The judicial power applies law held by the High Court held by the Supreme Court

=> Both the UK and the US respect the separation of powers: each power (legislative, executive and judicial) is held by a different player (individual or institution) separated to prevent dictatorship, abuse of power and respect the rule of law.

3. The balance of powers
In the UK, the 3 powers are unbalanced as the legislative is stronger than the executive. It means that the monarch is controlled by Parliament, that royal power is limited which defines a parliamentary monarchy.
In the US, the powers are balanced because the 3 powers are equally shared. Besides they mutually control/check one another. It is the principle of checks and balances.

4. The sovereignty of the people
The sovereignty of the people means that legitimate power comes from the people, that the nation is the basis of power
In the UK Parliament, the most powerful institution, represented the English nation (the nobility in the House of Lords and the people in the House of Commons).
In the USA the people was represented in all political institutions : US voters elected the President indirectly and Congress (Senators and Representatives) directly.

II. UK’s parliamentary monarchy and US republic today (+ docs 2 & 5)

1. Democracies
Democracy is the government of the people, by the people, for the people
In 1689 and 1787, the UK and the USA weren’t democracies because the franchise was restricted to wealthy men. Moreover, in the USA African-American (Black) slaves didn’t have the right to vote.
The UK became a democracy in 1928 when women got the vote. In the USA, African-Americans were emancipated by President Abraham Lincoln in 1865 but they were intimidated and had to fight to actually get the right to vote one century later in 1965.
Today both the UK and US are democracies as everyone votes thanks to universal franchise.

2. The Queen’s role
In the UK today, the prime minister (PM) is the head of government and holds the executive power, he or she governs the country. The monarch is still the head of state but no longer has any real/actual power: it is said that Queen Elizabeth II (the 2nd) reigns but doesn't rule.
The Queen, however, is a major symbol of stability (she's been in power far longer than any PM) and unity (she is above politics : she doesn't express any political opinion) for the British people. She is also the head of the Commonwealth, an association between Britain and its former colonies, which is why she is still queen of Canada, Australia and New Zealand (governed by their Prime Ministers).