Why does a neural network store more information than a molecule like dna, even if the dna had the same number of base-pairs as neurons or synapses in the network?

Answer: Imagine a very simple neural network, with four neurons. Each neuron is connected to three other neurons in this example. Therefore, the total number of neuron connections is going to be 12 in this example. We're comparing this to a DNA molecule that is 12 bp long. Effectively, we can say that the DNA molecule is able to store 12 bits of information. The information the neural network can store however is basically a function of how many different firing patterns there can be. It's easy to see how that's a lot bigger than 12.