Actually it is not completely clear which deep neural nets you are referring to but I guess you are referring to Dense, aka fully connected, networks. It depends on your data but for simplification, based on the answer here, neurons of the first hidden layer try to find lines for separating the data, and the neurons of the subsequent layers try to consider them simultaneously, I suggest reading the hole answer. As you increase the number of neurons in the first layer, you try to increase the number of lines for separation, The last shape of the link, and as you try to increase the number of e.g. the second layer, the number of convex shapes will increase. If you increase them too much you will over-fit your data.
Increasing the number of filters, neurons or kernels, in convolutional nets has another meaning. If you increase the number of filters, it means that you try to find more features that can represent your data better. In conv layers you actually try to find features of data. Although you are somehow doing same procedure in a dense net but because the neurons are less connected and they get updated similarly in conv nets, the features learned by conv nets are completely different from dense layers. for more information I recommend you taking a look at here.