There are substantial differences within as well as between the systems of parliamentary democracy and presidential democracy. To take two specific examples as patterns, some of the differences and similarities between parliamentary democracy as practiced in the United Kingdom and presidential democracy as practiced in the United States are as follows.
1. The president is Commander-in-Chief, head of both state and government. The prime minister in a parliamentary democracy is supposed to be "first among equals," the leading minister who remains responsible to the monarch, who is head of state. While not all parliamentary systems have a monarch, most grew out of systems which used to have one.
2. Anyone can run for president, subject to certain conditions, and the president is directly elected by the people (though, in the United States, this occurs via the electoral college). In a parliamentary democracy, people vote for local candidates representing a particular party, and this party then chooses its leader. The leader of the party that can command a parliamentary majority (either through his/her own party or through a coalition) becomes prime minister.
3. Though a prime minister sits in the legislature (as does the entire cabinet), while the president does not, similar considerations affect the power of both leaders. A president may fail to command a majority in the House or the Senate, meaning that s/he can accomplish very little in terms of legislation. A prime minister who leads a coalition, or who has a very small parliamentary majority, is in a similarly weak position.