A deeper investigation consists in isolating the auto-correlations to see whether the remaining values, once decorrelated, behave like white noise, or not. If departure from white noise is found, then it means that the time series in question exhibits unusual patterns not explained by trends, seasonality or auto correlations. This can be useful knowledge in some contexts such as high frequency trading, random number generation, cryptography or cyber-security. The analysis of decorrelated residuals can also help identify change points and instances of slope changes in time series.
Dealing with serial correlation is a big issue in econometrics; if you don’t deal with it in an Ordinary Least Squares regression, your regression will appear to have more explanatory power than it really does.