We present deep, wide-field Subaru Hyper Suprime-Cam photometry of two recently discovered satellites of the Milky Way (MW): Columba I (Col I) and Triangulum II (Tri II). The color-magnitude diagrams of both objects point to exclusively old and metal-poor stellar populations. We re-derive structural parameters and luminosities of these satellites, and find $M_{\rm V, ColI}=-4.2 \pm 0.2$ for Col I and $M_{\rm V, TriII}=-1.2 \pm 0.4$ for Tri II, with corresponding half-light radii of ${r}_{\rm h, ColI}=117 \pm 17$ pc and ${r}_{\rm h, TriII}=21 \pm 4$ pc. The properties of both systems are consistent with observed scaling relations for MW dwarf galaxies. Based on archival data, we derive upper limits on the neutral gas content of these dwarfs, and find that they lack H I, as do the majority of observed satellites within the MW virial radius. Neither satellite shows evidence of tidal stripping in the form of extensions or distortions in matched-filter stellar density maps or surface-density profiles. However, the smaller Tri II system is relatively metal-rich for its luminosity (compared to other MW satellites), possibly because it has been tidally stripped. Through a suite of orbit simulations, we show that Tri II is approaching pericenter of its eccentric orbit, a stage at which tidal debris is unlikely to be seen. In addition, we find that Tri II may be on its first infall into the MW, which helps explain its unique properties among MW dwarfs. Further evidence that Tri II is likely an ultra-faint dwarf comes from its stellar mass function, which is similar to those of other MW dwarfs.