The spin transfer torque magnetoresistive random access memory (STT-MRAM) is the leading candidate for spin-based memories. Nevertheless, the high write energy and read disturbance of the STT-MRAM motivated researchers to find other solutions. The spin Hall effect (SHE)-based MRAM is an alternative for the STT-MRAM, which also provides nonvolatility, zero leakage, and competitive area per bit, but with a lower write current. This paper focuses on a systematic performance analysis of these two proposed memory solutions. The SHE requires an external field to deterministically switch perpendicular magnetic anisotropy magnetic tunnel junction (MTJ). A previous experiment showed that the SHE can switch composite MTJ containing an in-plane layer without any field. In this paper, both traditional and composite MTJ structures are modeled in SPICE, which can reproduce realistic MTJ characteristics with user-defined input parameters. This self-contained model is used to compare the write energy and delay of the STT-MRAM and the SHE magnetoresistive random access memory (SHE-MRAM) for various write schemes including thermal fluctuation. Our simulations show, compared with the STT-MRAM, that the SHE-MRAM improves the write delay and the energy by eight times and seven times, respectively. Based on our extensive analysis incorporating the latest advances in magnetic materials and device technology, we predict that the SHE-MRAM is a feasible low-energy memory solution for future computing systems.