There are countless different flavors of Petri nets. The edges can be weighted, meaning that a transition can get more than one token from a given input place to fire, and can put more than one token in a output place when it fires.
About the choosing which edges they follow: You don't. In standard Petri nets firing is concurrent: If tokens can be used by more than one transition at the same time, they will non-deterministically go one way or another. You can actually refine this situation by extending your formalism, e.g. to timed nets.
I am not an expert of neural nets, but I'd guess they are more similar to signal flow graphs. These are related to Petri nets tho, but in a very deep and complicated way that I have no chance of explaining here right now. Check out the work of Sobocinski, Piedeleu and Zanasi about additive relations if you are interested in this!
About the choosing which edges they follow: You don't. In standard Petri nets firing is concurrent: If tokens can be used by more than one transition at the same time, they will non-deterministically go one way or another. You can actually refine this situation by extending your formalism, e.g. to timed nets.
I am not an expert of neural nets, but I'd guess they are more similar to signal flow graphs. These are related to Petri nets tho, but in a very deep and complicated way that I have no chance of explaining here right now. Check out the work of Sobocinski, Piedeleu and Zanasi about additive relations if you are interested in this!