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Based on what I have observed, disabling homotopy class planning makes planning much more vulnerable to being stuck with infeasible trajectories when trying to move in narrow spaces or otherwise very close to obstacles. But it does speed up local planning a lot when many obstacles are nearby. Performance tends to drop several times if there are many obstacle cells near the robot with homotopy enabled.
As far as I understand, TEB optimizes a trajectory locally, preferring to modify an already existing plan a bit rather than choosing a completely different one, even when an almost direct trajectory to the goal is available, or the chosen plan is very close to obstacles. When homotopy is enabled, there are several trajectories that are optimized at the same time, so there are more to choose from and the chosen trajectory can jump between very different routes as the result.
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