Photon counting detectors (PCDs) for x-ray computed tomography (CT) are the future of CT imaging. At present, semiconductor-based PCDs such as cadmium telluride (CdTe), cadmium zinc telluride, and silicon have been either used or investigated for clinical PCD CT. Unfortunately, all of them have the same major challenges, namely high cost and limited spectral signal-to-noise ratio (SNR). Recent studies showed that some high-quality scintillators, such as lanthanum bromide doped with cerium (LaBr3:Ce), are less expensive and almost as fast as CdTe. The objective of this study is to assess the performance of a LaBr3:Ce PCD for clinical x-ray CT. We performed Monte Carlo simulations and compared the performance of 3 mm thick LaBr3:Ce and 2 mm thick CdTe for PCD CT with x-rays at 120 kVp and 20-1000mA. The two PCDs were operated with either a threshold-subtract (TS) counting scheme or a direct energy binning (DB) counting scheme. The performance was assessed in terms of the accuracy of registered spectra, counting capability, and count-rate-dependent spectral imaging-task performance, for conventional CT imaging, water-bone material decomposition, and K-edge imaging with tungsten as the K-edge material. The performance for these imaging-tasks was quantified by nCRLB, that is, the Cramér-Rao lower bound on the variance of basis line-integral estimation, normalized by the corresponding value of CdTe at 20mA. The spectrum recorded by CdTe was distorted significantly due to charge sharing, whereas the spectra recorded by LaBr3:Ce better matched the incident spectrum. The dead time, estimated by fitting a paralyzable detector model to the count-rate curves, was 20.7, 15.0, 37.2, and 13.0ns for CdTe with TS, CdTe with DB, LaBr3:Ce with TS, and LaBr3:Ce with DB, respectively. Conventional CT imaging showed an adverse effect of reduced geometrical efficiency due to optical reflectors in LaBr3:Ce PCD. The nCRLBs (a lower value indicates a better SNR) for CdTe with TS, CdTe with DB, LaBr3:Ce with TS, LaBr3:Ce with DB, and the ideal PCD, were 1.00±0.01, 1.00±0.01, 1.18±0.02, 1.18±0.02, and 0.79±0.01, respectively, at 20mA. The nCRLBs for water-bone material decomposition, in the same order, were 1.00±0.02, 1.00±0.02, 0.85±0.02, 0.85±0.02, and 0.24±0.02, respectively, at 20mA; and 0.98±0.02, 0.98±0.02, 1.09±0.02, 0.83±0.02, and 0.24±0.02, respectively, at 1000mA. Finally, the nCRLBs for K-edge imaging, the most demanding task among the five, were 1.00±0.02, 1.00±0.02, 0.55±0.02, 0.55±0.02, and 0.13±0.02, respectively, at 20mA; and 2.45±0.02, 2.29±0.02, 3.12±0.02, 2.11±0.02, and 0.13±0.02, respectively, at 1,000mA. The Monte Carlo simulations showed that, compared to CdTe with either TS or DB, LaBr3:Ce with DB provided more accurate spectra, comparable or better counting capability, and superior spectral imaging-task performances, that is, water-bone material decomposition and K-edge imaging. CdTe had a better performance than LaBr3:Ce for the conventional CT imaging task due to its higher geometrical efficiency. LaBr3:Ce PCD with DB scheme may be an excellent alternative option for CdTe PCD.