Stan gives you the ability to do probabilistic reasoning. There is actually Tensorflow Probability (https://www.tensorflow.org/probability) which has a lot of overlapping algorithms, but isn't as mature and approaches some things differently.

The main difference is that with Stan you think in terms of random variables and distributions (and their transformations), while with Tensorflow/DL you think in terms of predicting directly from data. Stan lets model a problem with probabilities and do arbitrary inference, generally asking any question you want about your model.

There are many other interesting alternatives, e.g. http://pyro.ai/ which takes a yet another approach merging DL and probabilistic programming with variational inference. (Stan and TFP can do variational inference too, but I guess it's like Python vs JavaScript vs Ruby vs Java - all of them can be used for programming, but not the same way).