On a microscopic level, some cells in the leaf retain their ability to transform to numerous cell types, or in some cases revert to this ability. Given the proper conditions, these cells will differentiate into root and leaf parenchymal, gland, and dermal/epidermal tissue, etc.
On a macroscopic level, differentiation, growth and trophic factors (and extrinsic environmental factors) orchestrate the organization of the cells and, hence, its much of its visible characteristics.
There are many reasons why a cutting will die--most have to do with the fact that they are much more vunerable to unfavorable factors and much less able to adequately respond to them.