Luminosity is the total amount of energy produced by a star in a given time interval (e.g. ergs/s)
Flux is how much energy passes through a detector at a given location, or more specifically, flux is the amount of energy incident on an area in a specified amount of time.. This determines the apparent brightness of the source.
On astronomical scales, the Sun (or any star) can be taken to be a point source with a specific luminosity, or emitting a certain amount of energy over time. The energy expands out from the point in a spherical shell, so the emitted energy is spread over the surface area of a sphere of radius the distance from the star...the surface area is proportional to square of the distance from the star.
All of this reduces to the following relationship:
Flux = Luminosity / (4 * distance^2)
Flux = (4 * 10^33 ergs/s) / [(4)(1.5 * 10^13 cm)^2] = 4.4 * 10^6 ergs/s-cm^2