Computational differences between spiking neural networks and previous ANNs
Rate-based-network is a specific implementation of the more general spiking-networks. One may see a rate-based-network as a spiking-network in which the inputs from each neuron are accumulated over a short time-period (think of "one second") and are used to update the state of their target neuron only once in each time-period.

For example, if an input neuron has an output of '40' in a rate-based-network, you should think about it as if it fire 40 times in the time-period, and only once in each time-period it's target neuron 'read' this input.

On the other hand, with spiking-neurons the output is limited to binary and hence the time-period used must be shorter than the maximal rate. In each time-period the target neuron updates its state based on it's (binary) input. 

So to summarize:
* Spiking networks describe a more general phenomena.
* Spiking networks can react to inputs with fine temporal structure and act in time-periods which are order of magnitude smaller than rate-based-models.

Also one may add that spiking networks are likely to be more accurate biological description of the brain (but still a very rough account).