Is a directed graph different from a flow graph?

A directed graph like any mathematical object, is a well-defined object. It has a definition, so that handed any other mathematical object $X$, we can say whether or not $X$ is an example of a graph.

But what is the definition of a flow chart? Until you give a precise definition of flow chart, it's not a mathematical object and it's impossible to say how it compares to a directed graph. Furthermore, there isn't necessarily a correct definition of a flow chart; you get to define a flow chart however you need, depending on what you're thinking about. For example do you need to flow chart to "terminate," where flowing throw the chart you hit a terminal vertex? or can you have loops in your flow chart?