Let's assume that you are training a model whose performance depends on a set of hyperparameters. In the case of a neural network, these parameters may be for instance the learning rate or the number of training iterations.
Given a choice of hyperparameter values, you use the training set to train the model. But, how do you set the values for the hyperparameters? That's what the validation set is for. You can use it to evaluate the performance of your model for different combinations of hyperparameter values (e.g. by means of a grid search process) and keep the best trained model.
But, how does your selected model compares to other different models? Is your neural network performing better than, let's say, a random forest trained with the same combination of training/test data? You cannot compare based on the validation set, because that validation set was part of the fitting of your model. You used it to select the hyperparameter values!
The test set allows you to compare different models in an unbiased way, by basing your comparisons in data that were not use in any part of your training/hyperparameter selection process.