Although the two terms are often used interchangeably, a political party can be a faction, but a faction is not necessarily a political party. A political party exists for the purpose of obtaining and maintaining power within the government. A faction is a group within a group that maintains influence.
The modern definition of factions does not include James Madison's original definition that they are groups that exist for the purpose of limiting citizens' rights and well-being. Factions form within a political party, nation, corporation or virtually any other venue in which those who maintain power have control over crucial decisions and policies, usually as the result of discontent about the existing powers.
Early factions in the United States formed as the result of dissent about the government's role regarding the rights of citizens. One group of forefathers believed that the role of the federal government must be limited, and the individual states must be at liberty to make many decisions, while others believed a large federal government with limited states' rights was better. Factions of like-minded people formed on both sides of the debate. These factions became the basis for the political party system, which the Constitution was intended to prevent.