Both of them represent the different states a system can take. So what is the difference of Petri Nets and Finite State Machines? When do I use Petri Nets, and when do I use Finite State Machines?
Standard finite state machine contain only a single current state. Whereas in Petri nets multiple locations, more or less comparable with states in a finite state machine, can contain one or more tokens. A finite state machine is single threaded while a Petri net is concurrent.
In a finite state machine the active state changes in response to an event. In a Petri net transitions are executed as soon as all input locations contain at least one token.
A finite state machine can be considered as a special case of a Petri net.
In general I would recommend using a finite state machine if your process, or the part you wish to represent, is single threaded: fellow software engineers are probably more familiar with finite state machines; and there are more tools to convert a finite state machine to an implementation.
Use a Petri net only when you need the concurrency or extra expressivity. Or when you are modeling a factory plant where half fabricates are transformed into products or when your audience is more familiar with this image.
Perhaps Petri nets can also be used to model, visualize running, massive concurrent systems such as micro service architectures, azure service fabric reliable services and reliable actors, services running on kubernetus, azure function, and AWS Lambda.
In addition, there is more theoretical research about, and using, Petri nets than there is about finite state machines (note that, as I said earlier, finite state machines are reducible to Petri nets).