Why are MAVEN and Mars Orbiter Mission taking such different paths to Mars?

Two spacecraft launched for Mars this month: Mars Orbiter Mission on November 5, and MAVEN on November 18. MAVEN is now on an interplanetary trajectory, while Mars Orbiter Mission is still in Earth orbit and will not depart for Mars until the end of the month. A lot of people are asking me: why the difference? Here's your answer, with input from Dave Doody.