The components of the wide (73) separation quasar pair Q2345+007A,B (z = 2.15) have the most strikingly similar optical spectra seen to date (Steidel & Sargent) yet no detected lensing mass, making this system the best candidate known for a massive (~1014 M?) dark matter lens system. Here we present results from a 65 ks Chandra observation designed to investigate whether it is a binary quasar or a gravitational lens. We find no X-ray evidence for a lensing cluster to a (0.5-2 keV) flux limit of 2 ? 10-15 ergs cm-2 s-1, which is consistent with lensing only for a reduced baryon fraction. Using the Chandra X-ray observations of the quasars themselves, together with new and published optical measurements, we use the observed emission properties of the quasars for further tests between the lens and binary hypotheses. Assuming similar line-of-sight absorption to the images, we find that their X-ray continuum slopes are inconsistent (?A = 2.30 and ?B = 0.83) as are their X-ray-to-optical flux ratios. The probability that B suffers intrinsic absorption sufficient to account for these spectral differences is negligible. We present new optical evidence that the flux ratio of the pair is variable, so the time delay in a lens scenario could cause some of the discrepancies. However, adequately large variations in overall spectral energy distribution are rare in individual QSOs. All new evidence here weighs strongly toward the binary interpretation. Q2345+007 thus may represent the highest redshift example known of an interaction-triggered but as-yet unmerged luminous active galactic nucleus.