As a postdoc applicant, you should be starting to develop a distinctive, compelling research agenda and vision. As a grad student, it's usually enough to work on problems someone gives you, and you can go to your advisor for help as needed. If you never move beyond this, it's not a great outcome, but you'll eventually get your Ph.D.; you just won't be on track for further research success. To be a successful postdoc, you need to develop the ability to work more independently. However, intellectual leadership is more than just independence. It's not enough just to write papers. Instead, these papers should fit together and be aimed towards larger goals, in a way you can articulate coherently. If someone asks you what sort of research you do, you should be able to explain what you are aiming for, how your work differs from other people's, and why your approach is worthwhile and even important.
This takes time and comes only with experience - more will be expected of tenure-track job applicants, or people being considered for tenure or a full professorship. But a postdoc applicant needs to demonstrate a start, by writing a research statement that sounds exciting, and ideally by publishing several papers that build on each other to accomplish something larger than any one of them. It's even better if other researchers have already started to take up your ideas and apply or extend them, although that's more common in fast-moving fields with short publication cycles.
What you don't want is to come across just like your advisor, but with less experience and perhaps less talent. You can succeed that way if you do fantastic work, but it's not a wise approach for most students: it's just not good advertising to say "I'm basically a less good version of my advisor." Even while you are still in graduate school, you should be aiming to differentiate yourself a little from your advisor (with your advisor's help - after all, they shouldn't be trying to produce a clone). Find some important aspect of the subject that interests you more than your advisor, learn some useful background they don't know well, or find a collaborator who doesn't work with them.
As for how committees judge this, you've got to make a convincing case in your application. Your research statement should sound like a future leader, not a student (don't be arrogant, but you shouldn't sound like you are implementing someone else's vision - you need to demonstrate clearly that you understand how everything fits together and are adding your own ideas). You need to back your research statement up with publications worthy of it, as well as some concrete ideas for future work. Your letters of recommendation should convey the impact you are already having on the field.
Of course this is the best case scenario, and not every application will achieve this, but if you want to get a prestigious postdoc position and go on to a tenure-track job at a research university, then this is what you should be aiming for. You may feel like a fraud even trying to do this, since you know perfectly well that you still have much more to learn, but you should ignore these feelings. Everyone recognizes that postdoc applicants are still developing their research abilities, but it's important to be ambitious and show what you are capable of. (In fact, no matter how far you progress in your career, you'll never reach the point of being able to say "OK, I've reached my full potential and know everything I need to," but you can't let that stop you.)