A Validation set can be considered as a part of the training set as it is used for parameter selection and to avoid overfitting of the model being built.
On the other hand, a Test Set is used for testing or evaluating the performance of a trained machine learning model.
In simple terms, the differences can be summarized as; training set is to fit the parameters i.e. weights and test set is to assess the performance of the model i.e. evaluating the predictive power and generalization.