Yes, we need both. I think forecasting is more specifically about time-series data -- "What happens next?" Prediction the task of deploying a model to assess unseen data in some way. With prediction, there need not be any time-series component at all; you might just be interested in how well a model does against a holdout set of observations.
Support of this can be found in Cressie & Wikle Statistics for Spatio-Temporal Data, p. 17:
Uncertainty in data, processes or parameters means that there will be uncertainty in conclusions. Statisticians call this drawing of conclusions in the presence of uncertainty, statistical inference (or just inference); in this book, inferences will be either estimation of fixed but unknown parameters, or prediction of unknown random quantities. (Notice that "forecasting," namely concluding something about the future, is a special case of "prediction.")