The difference has to do with the amount of mass that was originally there in the star. Stars that have masses greater than around 5 times the mass of the Sun may end up as a black hole. In terms of observations, not many scientists believed that black holes could actually exists in nature. However, in 1963, Maarten Schmidt discovered Quasars. And the search for black holes was born.
Quasars are extremely bright, small objects.
They are considered bright since they are extremely distant from us, yet we can easily see them. They are considered to be small because of their rapid variability. One of the strongest arguements for a quasar is that it is a black hole that is actively feeding. This is based on measuring the energy output from the quasar and its small size. The idea is that as a galaxy forms, there is more material around for a black hole to eat.