We present multiple lines of evidence that gas in the Taffy galaxy system (UGC 12914/5) is in a highly disturbed state as a result of a recent head-on collision between these two gas-rich galaxies. Building on evidence of large quantities of warm molecular hydrogen in the Taffy bridge, and evidence of boosted [CII] emission from Herschel, we observed the Taffy system with ALMA in the CO 2-1 and 3-2 lines, and with the McDonald Observatory VIRUS-P IFU system in the visible light. Strong emission lines were detected from the galaxies and in the bridge with VIRUS-P, showing two dominant velocity components over much of the bridge and parts of both galaxies. We made a detailed analysis of the excitation properties of the ionized gas, finding that a significant amount of the high-velocity component of the bridge gas was consistent with shocks with velocities of 200-300 km/s. Two distinct filaments of ionized gas were found between the galaxies. Our ALMA observations also showed that the dense molecular gas is also highly disturbed. The most remarkable aspect of the ALMA observations is the discovery of a tangled web of molecular filaments in the bridge. The individual filaments are typically 1 kpc in length, are almost unresolved at 60 pc resolution in the other dimension. They also contain clumps that have broad line widths of 80-150 km/s. Although some of the filaments are associated with a prominent extragalactic star formation region located in the bridge, most of the filaments show little obvious sign of star formation despite their apparently high molecular surface densities. Given their high surface densities, it is not clear why the filaments are not forming stars, unless they are magnetically supported. One possibility is that they represent transient density enhancements in a supersonic turbulent multi-phase fluid created in the aftermath of the high-speed collision.