Neptune's Trojan asteroids have been observed to have a variety of optical colors, most notably red (g $-$ r < 0.75) vs. ultra-red (g $-$ r > 0.75), but the underlying cause of these different color classifications is unknown. Near-IR spectroscopy can be used as a probe of the surface composition of these objects, as broad ice bands for a variety of materials are present in the near-IR. Here, we present the first results of a spectroscopic survey of Neptune's Trojan asteroids using the NIRSpec instrument on JWST. We compare the near-IR spectra of eight Neptune Trojans (NTs) based on different optical color classifications and with model spectra of different ices. We find that most of our targets are consistent with a surface covered in a thin layer of H$_2$O and CO$_2$ ices, while the only NT to reliably be classified as ultra-red is covered in ice tholins in addition to CO$_2$. Ice tholins are a known reddening agent when subjected to irradiation, so these results support the hypothesis that differences in optical color are due to differences in irradiation of the surfaces of these bodies. Since NTs have very similar orbits and therefore generally similar levels of irradiation at the current time, our results suggest that these objects have unique origins or there is ongoing processing of the surfaces of these objects through stochastic disturbances such as impacts.