I would say predictions are often used in science e.g. If I mix these two chemicals together, I predict...will happen. So it's definite and unambiguous - "The weather forecast predicts rain." "I predict she will get drunk at the party" - I am SURE she will get drunk! "Anticipate" is more of a guess or a hunch "I anticipate rain" - I THINK it is going to rain. "I anticipate a bad/negative result." I hope the above is helpful. Maybe a TEFL teacher could explain it better.