One of the key differentiation between the two is granularity of the problem. Typically forecasting is done at a higher level - For example: “How much sales can we expect in next month?” or “What are the expected call volumes in a call center?”
Your aim is to understand the overall demand / supply and plan your business accordingly. Predictive modeling, on the other hand goes far granular - you ask questions like “Who are the customers who are likely to buy a product in next month?” and then act accordingly.
Of course, you can add the outcome of predictive model to get a forecast!
Hope this helps