The totally asymmetric simple exclusion process (TASEP) is a paradigmatic out-of-equilibrium transport model, with applications ranging from intracellular traffic to vehicular flow. When a localized defect (bottleneck) reduces the jump rate on one or more bonds, the system develops an alternative phase to the maximal current regime of the homogeneous model, and exact results become scarce. For this reason, numerical and analytical approximation methods are employed. This thesis provides a comparison between two families of approximate methods for the TASEP with a bottleneck: cluster approximations (Mean Field, Pair, and Triplet) and the Interacting Subsystem Approximation (ISA), showing how and in which cases these two families of approximations coincide and why.