With the recent availability of Noisy Intermediate-Scale Quantum devices, the potential of quantum computers to impact the field of combinatorial optimization lies in quantum variational and annealing-based methods. This paper further compares Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) in solving Higher Order Binary Optimization (HOBO) problems. This case study considers the hypergraph partitioning problem, which is used to generate custom HOBO problems. Our experiments show that D-Wave systems quickly reach limits solving dense HOBO problems. Although the QAOA demonstrates better performance on exact simulations, noisy simulations reveal that the gate error rate should remain under 10-5 to match D-Wave systems' performance, considering equal compilation overheads for both device.