DOE and regression use the same math for the analysis, and use the same diagnostics to check the residual assumptions. The key difference between DOE and generic regression is DOE involves control of the factors that generate changes in the responses. Regression is simply finding a mathematical relationship between variables. Ideally, in a DOE, one tries to keep the changes in the factors independent of each other. Regression methods without the benefit of design and forethought will have less power than a comparable controlled, designed experiment.
If you only have historical data and want to see CORRELATION, use Regression.
If you can change parameters and want to predict your process (by understanding CAUSATION), use DOE.