I work with highly seasonal data (city-wide water consumption) and Prophet has been a great tool for us.
In terms of performance, it has been the best for a few of our forecasts, compared to GRUs, LSTM, ARIMA and SARIMA. When it wasn't the best, it wasn't too far from the best model. But, to be fair, our forecast are of quite stable data, so most models do well.
However, I would say that the key strength of Prophet is how easy it is. You can produce results really fast, you can throw data with missing range, holidays, and it has interpretability components out of the box. It depends on what do you need, but for most of our tasks, we and our stakeholders are more than happy to sacrifice a bit of performance for this features.
Does it apply any kind of persistence/memory on the instantaneous exogenous variables you feed it? E.g. if you feed it the exogenous variable of "temperature right now", is it able to create a new exogenous feature "average temperature over the last three time steps"?
No, you have to handcraft all (transformations) of exogenous features. But since it's really all linear regression, that's usually reasonably straightforward.
In terms of performance, it has been the best for a few of our forecasts, compared to GRUs, LSTM, ARIMA and SARIMA. When it wasn't the best, it wasn't too far from the best model. But, to be fair, our forecast are of quite stable data, so most models do well.
However, I would say that the key strength of Prophet is how easy it is. You can produce results really fast, you can throw data with missing range, holidays, and it has interpretability components out of the box. It depends on what do you need, but for most of our tasks, we and our stakeholders are more than happy to sacrifice a bit of performance for this features.