Other answers have gone over the basics of what each thing is. At the higher level, a flowchart is a design level tool, while DFDs are more analysis.
DFDs have some nice features. Since they show the flow of data, some things become more obvious when charted this way: some data is only used by a few routines, some routines use only some bits of data, some routines touch everything. Seeing that up front helps organize, restructuring, and planning.
A follow-on worth exploring is the Event-Response Diagram, which is basically a DFD only showing process and data needed to process an "event", meaning something triggered externally (customer makes payment, etc.).