Assigning vector of weights to Neural Network
From the documentation we can find out that:


weights: (case) weights for each example – if missing defaults to 1.
Wts: initial parameter vector. If missing chosen at random.


Though I'm no R expert, my understanding is:


weights is something like sample_weight in python, which controls the weight of each sample. It's useful when dealing with imbalanced class issue.
Wts is the initial weights of neural network. Random weights will be initiated if this parameter is not passed.