Why could we use DoG? An old and low-cost edge-enhancement technique consists in subtracting a blurred image from the original image. Its filter interpretation is an impulse filter (the neutral filter for the original image) minus the blur filter. The resulting filter is positive at its center, and negative around, a sort of coarse Laplacian filter.
The above impulse filter can be seen as the limit of a Gaussian filter whose $\sigma$ tends to $0$.
The DoG filter somehow generalizes the described old edge-enhancement technique, with a pair of blurring filters. Perhaps surprisingly, two blurs, when subtracted, can sharpen edges. The resulting filter acts as a pass-band with appropriate $\sigma_k$ kernel ratios.
How effective is DoG? its effectiveness is close to that of the Laplacian of Gaussians, which it mimics, see below in 1D.
Which type of images work best for this filter? It is believed to work well for large images, since it has very fast implementations. Due to the "double filtering effect" in high frequencies, it is somewhat robust to noise. Depending on the kernel size choice, it enhances features that possess a certain scale extend.