What tasks you train with one set of features and predict with another? The most common scenario in supervised learning is to have data points with a set of features and train a model to make classification predictions afterward.

Usually, for predictions to make sense with new data points, these new data points need to have the same features and come from the same distributions.

My questions is: what are scenarios in which the data points you want to predict do not necessarily have the same set of features from the data points you trained your model with?
If you mean completely new set of features for prediction: it will not be helpful. Your model 'learns' something on the training feature space and you hope to apply the learning to new data-points in the same feature space. If the feature space for prediction is completely new, the learning will be useless!
If you mean somewhat new set of features: that often happens in text applications where new tokens might show up in the data which were never seen in training data. Generally handled by some strategy like zero vector or random vector.
Same features but different distribution: this can happen in practice but model performance will go down.