The way I understand it, the key difference is that when training o1, they were going beyond simply "think step-by-step" in that they were feeding the "step-by-step" reasoning patterns that ended up with a correct answer back into the training set, meaning the model was not so much trained to find the correct answer directly, but rather to reason using patterns that would generally lead to a correct answer.
Furthermore, o1 is able to ignore (or even leverage) previous reasoning steps that do NOT lead to the correct answer to narrow down the search space, and then try again at inference time until it finds an answer that it's confident is correct.
This (probably combined with some secret sauce to make this process more efficient) allows it to optimize how it navigates the search space of logical problems, basically the same way AlphaZero navigated to search space of games like Go and Chess.
This has the potential to teach it to reason in ways that go beyond just creating a perfect fit to the training set. If the reasoning process itself becomes good enough, it may become capable of solving reasoning problems that are beyond most or even all humans, and in a fraction of the time.
It still seems that o1 still has a way to go when it comes to it's World Model. That part may require more work on video/text/sound/embodiement (real or virtual). But for abstract problems, o1 may indeed be a very significant breakthrough, taking it beyond what we typically think of as an LLM.