My understanding is that deep learning has the largest benefit with perceptual data (e.g., images, speech, and to a lesser extent natural language and control). The driving force behind deep learning is the ability to "compress" data by learning hierarchical representations. For example, your images of cats and dogs are very high dimensional and encode a lot of redundant information, making them especially suitable for such compression.
On the other hand, deep nets probably won't outperform more traditional machine learning algorithms on less high dimensional, uncompressed data. This problem is exacerbated by the amount of tuning needed by deep nets. Architectures have been fine tuned for common data types such as images and speech, but if the dataset doesn't fall into one of these categories, you have to tune the algorithms yourself.
On the other hand, deep nets probably won't outperform more traditional machine learning algorithms on less high dimensional, uncompressed data. This problem is exacerbated by the amount of tuning needed by deep nets. Architectures have been fine tuned for common data types such as images and speech, but if the dataset doesn't fall into one of these categories, you have to tune the algorithms yourself.