Short answer:
Strictly speaking, "Deep" and "Spiking" refer to two different aspects of a neural network: "Spiking" refers to the activation of individual neurons, while "Deep" refers to the overall network architecture. Thus in principle there is nothing contradictory about a spiking, deep neural network (in fact, the brain is arguably such a system).
However, in practice the current approaches to DL and SNN don't work well together. Specifically, Deep Learning as currently practiced typically relies on a differentiable activation function and thus doesn't handle discrete spike trains well.
Further details:
Real neurons communicate via discrete spikes of voltage. When building hardware, spiking has some advantages in power consumption, and you can route spikes like data packets (Address Event Representation or AER) to emulate the connectivity found in the brain. However, spiking is a noisy process; generally a single spike doesn't mean much, so it is common in software to abstract away the spiking details and model a single scalar spike rate. This simplifies a lot of things , especially if your goal is machine learning and not biological modeling.
The key idea of Deep Learning is to have multiple layers of neurons, with each layer learning increasingly-complex features based on the previous layer. 716
For example, in a vision setting, the lowest level learns simple patterns like lines and edges, the next layer may learn compositions of the lines and edges (corners and curves), the next layer may learn simple shapes, and so on up the hierarchy. Upper levels then learn complex categories (people, cats, cars) or even specific instances (your boss, your cat, the batmobile). One advantage of this is that the lowest-level features are generic enough to apply to lots of situations while the upper levels can get very specific.
The canonical way to train spiking networks is some form of Spike Timing Dependent Plasticity (STDP), which locally reinforces connections based on correlated activity. The canonical way to train a Deep Neural Network is some form of gradient descent back-propagation, which adjusts all weights based on the global behavior of the network. Gradient descent has problems with non-differentiable activation functions (like discrete stochastic spikes).
If you don't care about learning, it should be easier to combine the approaches. One could presumably take a pre-trained deep network and implement just the feed-forward part (no further learning) as a spiking neural net (perhaps to put it on a chip). The resulting chip would not learn from new data but should implement whatever function the original network had been trained to do. 716