I think OP means vector notation in the research papers, not using vector operations and representations at the code level.
That said, it can be useful for the beginner to implement a basic NN library from the ground up, to understand how the vector processing works, as well as what is going on step-wise with backprop and such.
Once that is fully understood, the next step of utilizing true vector processing libraries can be taken, and so on - eventually culminating in using and understanding libraries like TensorFlow.
Having the background of the lower levels gives you an appreciation and even insights when you transition to higher level frameworks.
That said, it can be useful for the beginner to implement a basic NN library from the ground up, to understand how the vector processing works, as well as what is going on step-wise with backprop and such.
Once that is fully understood, the next step of utilizing true vector processing libraries can be taken, and so on - eventually culminating in using and understanding libraries like TensorFlow.
Having the background of the lower levels gives you an appreciation and even insights when you transition to higher level frameworks.
That's just my opinion, though.