A budget sets out the plan for what a business wants to achieve; a forecast states a business’s actual expected for results,.
A budget is a quantified plan; a forecast is what is expected to happen.
At the start of the year these will probably be identical. As the year progresses they are likely to diverge so that towards the end of the year budgeted sales for the year might be 100,000 (the original figure) but forecast sales might be only 80,000.