Forecasting tries to answer questions like "can we predict the distribution of values of a time series variable at some point in the future?" Consider Sugihara's simplex projection (a kind of state space reconstruction method), which can make reasonable short term forecasts on a time series variable, even when that variable is itself causally linked with other unmeasured variables.
Time series analysis may instead ask questions like "what explains the behavior of variables across time?" Consider Abadie's synthetic control methods as ways of explaining the causal effect of policies on macro-level variables.
Your question gets at the distinctions between explanation and prediction. Really good explanations may not give much predictive power. Really good predictive systems, may even behave as a black box, and provide little to no explanation. In a time series context explanation and prediction are also both domains of concern about uncertainty and inference.
Finally, I would say that, my pointing at distinctions between prediction and explanation aside, "time series analysis" is a broad term, and covers explanatory methods, some people would probably see forecasting as a subset of time series methods, and of course, some people will simply be interested in the behavior of a time series (for example in an AR(1) setting, drawing distinctions between strong and weak stationarity and unit root, moving average errors, behavior of expected values, etc.).
Summarizing, I would say that "time series analysis" broadly encompasses:
- Description and categorization of time-series behaviors
- Prediction of time-series behavior
- Explanation of time-series behavior
I read Hyndman as lauding the attention to #2 that the forecasting competitions were producing, including the critical insight that explanatory (but perhaps also descriptive) time-series models do not necessarily produce wonderful predictions.
Abadie, A. (2021). Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects. Journal of Economic Literature, 59(2), 391–425.
Rescher, N. (1958). On Prediction and Explanation. The British Journal for the Philosophy of Science, 8(32), 281–290.
Scheffler, I. (1957). Explanation, Prediction, and Abstraction. The British Journal for the Philosophy of Science, 7(28), 293–309.
Shmueli, G. (2010). To Explain or to Predict? Statistical Science, 25(3), 289–310.
Sugihara, G., & May, R. M. (1990). Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature, 344(6268), 734–741.