This work presents a comparative study between the Topology Optimization of Binary Structures (TOBS) method and the Floating Projection Topology Optimization (FPTO) method applied to the design of foundation structures. TOBS employs Sequential Linear Integer Programming (SILP) to solve the topology optimization problem using binary variables, while FPTO relies on an implicit floating projection scheme to solve the relaxed continuous version of the optimization problem. Both methods are gradient-based approaches and are considered state-of-the-art techniques. The objective of this study is to evaluate the performance of the two methods in terms of structural efficiency, topological features, and computational behavior. In addition, the influence of different constitutive models for the soil-foundation interaction is investigated and compared within each optimization framework. Numerical models are developed for representative foundation domains under different loading and boundary conditions. The optimized designs are assessed based on structural compliance, material distribution, and robustness of the resulting topologies. The comparison reveals the advantages and limitations of each method when applied to structural foundation designs, offering insights into their suitability for practical engineering applications.