The order does matter. AOs only have so much time to spend and they will be scanning both sections quickly.
Start with the ECs that show they matter to the student through some combination of greater leadership/sustained involvement/hours spent/quantifiable impact. As involvement increases, those activities should move down the list.
In general, the activities where the student spends his or her time will be higher on the list, but it’s not absolute. One of my daughters spent the most time at the two sports she played, but listed an activity that she was really passionate about first. She had four years of participation and leadership in it, so it wasn’t just something she had done a couple of times, but it also wasn’t as time consuming as the sports. It made sense to the story she was telling about herself to list it first. My other daughter, though, did list first the activity where she spent the most time.
Similarly, with honors/awards, start with those that are more selective and/or more important to/representative of the applicant. Which brings him or her the most pride? Start from there.