Is there any significant difference between ～ないと、～なきゃ、なくちゃ when they mean 'must'?
～ないと is usually used for rules, social conventions, very important appointments, and is often abbreviated to just ～ないと. (with no explanatory clause following)
運動をしないとね = Undou wo shinai to ne / 'Cause if you don't work out, then... = "gotta work out"
手を洗わないとね = gotta wash your hands. (to me personally it feels more like "yeah because if you don't wash your hands then...")
～なきゃ feels colloquial (and in some cases hurried)
～なくちゃ feels colloquial
～なくてはいけません (roughly: must do this at all costs")