In addition to presidential and parliamentary systems having different processes for selecting executives (i.e., presidents and prime ministers), the two systems also have very different electoral systems for determining who serves in the legislature. In the United States presidential system, the legislature is broken up into congressional districts (for the House of Representatives) and states (for the Senate). In order to win a seat in the legislature, candidates must win a plurality (more than all other candidates) to win that seat in Congress. The candidate with the most votes get all the power of representing that district or state. In most parliamentary systems of government (there is a lot of variation around the world), legislatures are determined by a process called "proportional representation." Seats in parliament are allotted based on the proportion of the overall vote. If party X receives 30% of the overall vote, party X receives 30% of the seats in parliament. Therefore, another importance distinction between presidential and parliamentary systems is that voters cast ballots for parties in parliamentary systems, while voters cast ballots for candidates in presidential systems.
There are many differences between written and unwritten constitutions. Republics with unwritten constitutions typically consider all statutory law to be constitutional, meaning that all laws passed by legislatures or made by courts and judges are supreme. In republics with written constitutions, however, laws have varying degrees of importance and authoritativeness. For example, in the United States, the US Constitution is the "supreme law of the land," trumping all state constitutions, state and federal statutory law. A law passed by Congress can be considered unconstitutional if the Supreme Court determines that the statute violates the Constitution in some way. In republics with unwritten constitutions, no such tension or conflict exists: statutes passed by legislatures have the full weight of law and all concomitant authority.
Finally, written constitutions tend to be difficult to amend or change. The US Constitution has only been amended/modified 27 times in American history. Other republics, like the German Federation, have amended their written constitution hundreds of times. Republics with unwritten constitutions have far less cumbersome processes for modifying existing political institutions.
This question is asking about two separate ideas that are not necessarily connected to one another. The difference between a parliamentary system and a presidential system is not the same as the difference between a written and an unwritten constitution. I will compare and contrast the parliamentary and presidential systems in this answer.
The major similarity between these two systems is that they have elected legislative branches that make the laws for a given country. This is a very important similarity because it means that both systems are essentially democratic. They clearly differ in important ways, but they are fundamentally democratic.
Perhaps the most important difference between these two systems is the fact that a parliamentary system lacks an independent executive branch. In a parliamentary system, all of the leaders of the executive branch come from the legislative branch. This is in contrast to a presidential system. In a presidential system the two branches are completely separate and members of the legislature cannot also be members of the executive branch. In the US, for example, this is stated in Article I, Section 6 of the Constitution.
The other very important difference between these two is that the presidential system allows different parties to be in control of different parts of the government. In the parliamentary system, the legislative branch controls the executive branch. This means that the two branches can never be controlled by different parties. In the presidential system, by contrast, the president and the legislature are elected separately and independently. It is possible, as in the United States today, for the president to be of a different political party than the majority of the legislature. This cannot happen in a parliamentary system.
Thus, both of these systems are similar in a fundamental way because they are both democratic. However, they differ in important ways because of the lack of separation of powers in the parliamentary system.
For much more on this topic, please follow the link below.