I would say that the Original cast, particularly Captain Kirk, seemed a lot more confident. Even when faced with very difficult decisions, they at least seemed to take solace in the fact that they were always trying their best. I suppose, if you always give your best effort, you needn't fear failure because any failure will only result from things you were never able to control in the first place.
OTOH, the Next Generation crew seemed more self-conscious, more afraid of making mistakes. Perhaps it's because the stakes for them often felt more removed. Without the constant spectre of Cold War with the Klingons, it rarely felt like anything the crew did or didn't do would really affect anyone that we cared about. Even the ship & crew rarely seemed to be in any real danger because it felt less like a functional military vessel and more like an omnipotent space hotel.