a computational graph is made of node where is done "operation" on incoming variables. (see first paragraph from the link done in OP)
A neural network use perceptron or "neuron" model for each node. a generic example of neuron model is : Each incoming value is multiply by a "synaptic" weight, then are sum and the result will be pass to an activation function. Which will give an output result that could be project to next nodes. Neural Network can "learn" by adapting synaptic weights thanks to methods such as backpropagation or direct feedback alignment.
Any computational graph where nodes are not a kind of neuron/perceptron model are not neural network.