Parliamentary government and presidential government are similar in many ways. They are representative governments (republics) where usually all or part of the legislature is chosen democratically. They also both include an executive function. However, it is with the structure and power of the executive that they differ most.
In a parliamentary government, the executive, known as the prime minister (PM), is chosen by the houses of parliament from the members of the parliament. The PM is usually a member of the majority party in the parliament, which means that rarely is the executive of a different party than the majority of the parliament.
A PM remains as the executive until elections bring about a different majority in the parliament, unless he or she steps down or a no confidence vote is held. A successful no confidence vote in the parliament will dissolve the current executive government (PM and the cabinet) and require a new PM be elected form the parliament.
Finally the PM is not the head of state. The PM serves many of the executive functions a head of state would serve, but the PM is merely a member of the parliament chosen to perform the executive functions. Constitutional monarchies are usually parliamentary governments, where the monarch is the titular head of state while the PM performs the actual executive functions of the government.
In a presidential government, the executive and the legislature are different branches of the government. The president, who is the head of state as well as the chief executive, is elected separately from the election of the members of the legislature. Thus, in a presidential system, the executive can be from a different party from the majority party in the legislature.
Presidents are usually elected for a specific term and are unaffected by changes of the majority party in the legislature while the president is in office. Further, since presidents serve in a separate branch of government and hold a specific term of office, the legislature cannot remove a president via a no confidence vote.
These two systems could be diagramed effectively in a variety of ways. One would be comparative organizational charts. The parliamentary system would be a single tree with the PM as a component of the parliament. The presidential system would constitute separate trees for the separate branches.