We have searched for optical identifications for 79 Chandra X-ray sources that lie within the half-mass radius of the nearby, core-collapsed globular cluster NGC 6397, using deep Hubble Space Telescope Advanced Camera for Surveys Wide Field Channel imaging in Hα, R, and B. Photometry of these images allows us to classify candidate counterparts based on color-magnitude diagram location. In addition to recovering nine previously detected cataclysmic variables (CVs), we have identified six additional faint CV candidates, a total of 42 active binaries (ABs), two millisecond pulsars, one candidate active galactic nucleus, and one candidate interacting galaxy pair. Of the 79 sources, 69 have a plausible optical counterpart. The 15 likely and possible CVs in NGC 6397 mostly fall into two groups: a brighter group of six for which the optical emission is dominated by contributions from the secondary and accretion disk and a fainter group of seven for which the white dwarf dominates the optical emission. There are two possible transitional objects that lie between these groups. The faintest CVs likely lie near the minimum of the CV period distribution, where an accumulation is expected. The spatial distribution of the brighter CVs is much more centrally concentrated than those of the fainter CVs and the ABs. This may represent the result of an evolutionary process in which CVs are produced by dynamical interactions, such as exchange reactions, near the cluster center and are scattered to larger orbital radii, over their lifetimes, as they age and become fainter.