I am looking at this model, which is used when the residuals of your typical least squares regression model is serially correlated. https://online.stat.psu.edu/stat510/lesson/8/8.1. I think on google, it is also called regression plus time series errors.
It seems like the resulting coefficients of the regression + arima model is pretty much the same as just your original regression model. As a result, is there a reason to even take this extra step to rebuild your model jointly with an ARIMA model for the residuals - since your coefficients are pretty much the same so I'd imagine prediction and inference to be the same as well?