At the moment, the main choices are Tensorflow and probably Pytorch. Theano seems to be falling in popularity lately. I don't know much about Pytorch, but tensorflow also has libraries that sit on top of it like Keras or Tflearn which make development much quicker and easier. Your best bet is to do research on your own for both, and then try both, and see which fits your goals better.

Almost all AI libraries are going to be the same or very similar, because neural networks are all the same structures. TensorFlow is impressive to me because they have a lot of more advanced packages written for you already, whereas some of the other libraries can do basic neural networks for you with ease, but then struggle with complex networks or principles. For example, a lot of the research that Google does is done in TensorFlow, and they tend to release it open sourced in TF code. You can use it immediately, and that's cool. To use it in Pytorch or some other framework, you'd first need to do some converting, which you may or may not be able to do yourself.