What are the main differences between the parliamentary system of government versus the presidential system? For example, Germany's parliamentary system versus Mexico's presidential system. I'm particularly interested in the pros and cons of each.
The major difference between these two systems is that in a Presidential system, the executive leader, the President, is directly voted upon by the people (Or via a body elected specifically for the purpose of electing the president, and no other purpose), and the executive leader of the Parliamentary system, the Prime Minister, is elected from the legislative branch directly.
In the Presidential System, it is more difficult to enact legislation, especially in the event that the President has different beliefs than the legislative body. The President only responds to the people, the legislative branch can't really do anything to threaten the President. As a result, he can make it more difficult for the legislative body to do anything.
In the Parliamentary system, if the Parliament doesn't like the Prime Minister, they can cast a vote of no confidence and replace him. This tends to make the executive leader subservient to the Parliament.
Bottom line is, if you believe that government should have more checks and balances, then a Presidential system will give you that. If you believe that it should have the power to enact laws quickly, then you should go for a Parliamentary system.
Every system is different. The salient distinction between the two classes of systems is that, in a presidential system, executive power is constitutionally vested in a single individual (i.e., the president), whereas, in a parliamentary system, executive power is vested in the legislature (i.e., parliament, which chooses a prime minister or chancellor to head the executive).
Each system has its own advantages and disadvantages. Both forms of democracy are common, worldwide.
The main advantage of a presidential system is that it yields a strong and clear executive. This is important, especially in times of crisis or war. In contrast, parliamentary systems often result in a fragile or even absent executive. For example, Belgium was once left without an executive for 541 days because there were many parties and no majority could agree. Israel frequently calls early elections due to the governing coalition collapsing. Another example is Australia, where, as of 2018, no prime minister has lasted a full term in office since 2007, because the parties themselves have been consumed with infighting.
The main advantage of a parliamentary system is that the executive is accountable to the parliament. In most parliamentary systems, the prime minister is a member of parliament and must appear and answer questions from other members on a weekly basis. More importantly, the parliament may remove the prime minister at any time by a simple majority vote. (Compare that to the arduous process of impeachment.) This happens quite frequently, but also the mere threat of losing support forces the prime minister to consult with members of their party/coalition before making decisions.
In a presidential system, the cabinet serves at the pleasure of the president and has little independent authority. In a parliamentary system, the cabinet often contains rivals to the prime minister who are there because of the support they have in the party/coalition, rather than because the prime minister necessarily wants them. Indeed, in a parliamentary system, it is often the cabinet that makes decisions collectively, rather than the president making them unilaterally.
In both systems you need checks and balances to ensure that the executive is not too powerful. A parliamentary system automatically includes the parliament as a check. It usually only takes a small fraction of the governing party/coalition to cross the floor and bring down the government. A presidential system needs additional checks and balances to be codified -- usually in the form of a strong judiciary, the legislature controlling the supply of money, and, of course, the possibility of impeachment and removal from office in extreme cases.
Note that most parliamentary systems still have a separate president or constitutional monarch. The president/monarch is usually a ceremonial figurehead (i.e., head of state, with the prime minister being the head of government) with some constitutional and non-political functions. However, in some cases the president may be actively involved in politics, yielding a semi-presidential system.