I definitely agree, that Causality is a very interesting topic! And I even think its not that complex to think of it and understand how its implemented in the brain.
I suggest reading: Causal Inference in Statistics: A Primer (by Judea Pearl)
In the first chapter already, the difference is explained: Usually in statistics, there is data, that was collected. For example the two random variables C (for Cancer) and S (for Smoking) can take two values, and the data can be summarized by a table with 4 entries, showing the joint probabilities.
Based on this data, it is impossible to figure out, if Cancer causes Smoking, or Smoking causes Cancer.
We need something else to decide this question!
In the book, the author argues, that a directed graph is needed in order to solve the question, so it is known beforehand, what is a source and what is a effect.
However coming to the brain, the notion of time is very important, since when you first smoke, and then get cancer, it is pretty obvious which is the source and which is the effect! The brain therefore learns to build a causal model, that links from sources to their effects. And further I would argue, that is also why axons transmit signals in one direction: This way it is clear, what caused what, it is inherent to the model.