There are two bandwidth specs, full-power (large-signal) bandwidth, and small signal bandwidth. They are related by the structure of the amplifier, but the small signal bandwidth is normally much wider. The full-power bandwidth will be limited by the slew rate because the amplifier can't follow the fast rise of the larger amplitude sine wave. The slew rate is what matters in the step response because that's how long it takes to follow the input step. However, the settling time after slewing to the new point is governed more by the small-signal bandwidth (more is better). Amplifiers with the same slew rate but a different small-signal bandwidth will likely have somewhat different responses to a step input.
