One of my projects several years back ran both a LR model and a DNN against the same input data (albeit featurized differently). Accuracy, P&R were roughly the same (minor differences depending on the time horizon), but the LR model took maybe a half hour to train and five minutes to run; the DNN took about 24 hours to train and an hour or two to run.
This wasn't even particularly huge data compared to my other projects. But certainly at that scale, there are huge differences between regression & NNs.