This question is very broad, so let me attempt to answer it using my own background in time series analysis.
As an example, why would I continue using ARIMA to forecast a time series? Why not simply use an LSTM model by default, since this is a type of recurrent neural network that takes time-related dependencies into account?
Well, an LSTM model is not good at modelling all time series. It is effective when it comes to modelling volatile data, but ARIMA still outperforms when it comes to forecasting trend data - LSTM tends to overemphasise volatile patterns in future predictions.
Let's take an example of forecasting weekly hotel cancellations by potential customers. The second time series shows much more variability in the number of weekly hotel cancellations than the first:
H1 Time Series
H2 Time Series
Based on MDA (mean directional accuracy), RMSE (root mean squared error), and MFE (mean forecast error) - ARIMA demonstrates superior performance overall for the first time series, while LSTM shows better performance for the second:
On the basis of this example - which is quite specific given the broadness of your question - deep learning techniques are not always used because simpler models can perform better under certain circumstances. It is all about understanding the data you are working with and then choosing the model - not the other way around.