Based on the standard definitions of these words, they both relate to how these words are used in management. Budgeting in the real world refers to planning out how you will spend your money usually during the month so that you don’t run out and you will have everything you need to survive during that month. Forecasting is using the information that you have gathered and what plans you will make for the future.
The same can be said for these words regarding management. They refer to doing these things for a company. Budgeting has to do with money, but forecasting has to do with the future. Budgeting is usually done for a shorter period than predicting.