As a signal is by definition a time series, there is significant overlap between the two.
I would expect a book on time-series analysis to be either a mathematical treatment, or a business/commercial treatment, while a book on statistical signal processing is likely to make heavy use of mathematics, but interested in the problems of signal analysis, classification, noise reduction, and other problems relevant to engineering / applied science.
Statistical signal processing uses the language and techniques of mathematical time-series analysis, but also introduces into the problem domain many concepts and techniques out of electrical engineering: signal to noise, dynamic range, and time/frequency domain transforms.
In my view, time-series analysis is a mathematical field, which then has applications wherever time series tend to crop up. Those fields then develop techniques that are specialised for those problem domains, with a specialised body of knowledge.
As time series arise in business and economics, there is an industry of material on time-series forecasting, trend analysis, etc. Much of this 'commercial' application is not present in the material on statistical signal processing, in part because the nature of the two time series is very different: signals are continuous over both time and measurement variables (e.g. voltage, intensity, etc.) Whereas most business time-series are taken over a discrete time domain (days, weeks, months, quarters, years).