The initial motivation of a sparse representation for a class of signals or images is to pre-separate components with different morphological behaviors.
A traditional model for images is the augmented cartoon: piece-wise smooth + contours + geometrical textures + "noise (unmodelled)".
The mother wavelets (gradient-like operators) can capture isolated singularities at different scale with a few coefficients (and if they oscillate, a bit of textures as well). If the singularities are mixed with a polynomial trend, a standard gradient may also produce coefficients. With a sufficient number of vanishing moments, wavelets somehow "pre-separate" polynomials and singularities, and the polynomial part is cast, subsampled, to a small set of approximmation coefficients.
All in all, you can get few coefficients for singularities, and few coefficients for polynomials between singularities (a long polynomial can be encoded into a few polynomial factors). But still you preserve all the information, especially about the gradients you want to keep.
Wavelets are, under some theoretical conditions, quite optimal for such 1D signals. In practice on 2D data, other aspects come into play and counterbalance the need for vanishing moments: the nature of singularities in images, their finite size, the other wavelet properties, etc.
Resultingly, 1D wavelets work ok for not-too-complex images, but become limited with complicated data, complicated processing, or high noise conditions.
My experience is that, unless you have specific needs, 2-4 moments suffice in most of the cases, meaning that if finally you use a wavelet with more moments, it will be a by-product of other wavelet properties, and not a strong requirement. There are many options in finding the optimal multiscale geometric representation.