The benefits of deep learning have more to do with the number of features than the size of the dataset; e.g when you are dealing with million pixel images, you need a deep net to extract useful higher level features automatically. From there, yes, more data is better, but a better post in this vein would be, "Don't use deep learning your data doesn't have that many features".
I'd say the benefits come from the information content of the features or lack thereof. When you have uninformative features like pixel colors or word identities, there's nothing for traditional methods to work with. You have to start with feature engineering and pruning before decision trees or linear classifiers have a chance.
Most of the wins under the "deep learning" umbrella come from extracting meaning from homogenous features like "the pixel at x-2,y+1 has red=123" or "the word at n+1 is 'king'". That's why we see latent variable embeddings like word2vec come from the DL world even though they're not deep.
When you want to include highly informative features in a deep network, it's often better to feed them into a separate logistic model, as shown in the Tensorflow wide-deep tutorials.
Hmm I would refrain from saying DL is useful only when approaching 1 megapixel images.
State of the art performance on MNIST is held by a 6 layer convnet (4 layers convolutional, 2 layers FC). MNIST is just 28 x 28 grayscale images, so 768 dimensions. There are many more datasets on the same order of dimensionality. CIFAR 10/100 (32 x 32 pixel images) is also dominated by DL convnets, AFAIK.
Sure, didn't mean to imply megapixel images were the lower threshold, just that it is more to do with the number of features and the need to automatically extract higher level features.
