The dictionary suggests that they are synonyms but in reality they are used in different situations.
A forecast is a prediction of what is likely to happen in the future. It can used to talk about any situation but not medical situations.
A prognosis is a prediction of what is likely to happen with an illness or disease.
- I will present my sales forecast at the meeting on Friday.
- The weather forecast for the weekend isn’t good.
- He looks very ill at the moment, but the prognosis is good.
