Achieving high quality results can require networks with tens of thousands of weights, so processing an image using such a network is very slow.
Training is very slow, requiring many passes through the training data.
The complexity of processing a single training sample is linear in the number of network weights. A high-quality result might require tens of thousands of weights, and training sets can contain millions of samples. Unlike polynomials, neural networks do not have any guaranteed rate of convergence. Although convergence to the optimal F might be guaranteed in the limit as the number of weights approaches infinity, the rate of convergence might be slow enough to make such a guarantee meaningless.