Both involve letting go of all earthly tethers, seems pretty cut & dry to me. Even so, the fact that nobody had attained flight for centuries suggests that Aang would not. Most likely the difference is that firstly Zaheer was aware of the technique to attempt it & secondly flight requires an ongoing state of detachment, but Aang only detaches himself while using the Avatar State.
Basically? Flight requires one to be both an airbender and (as Neo stated) detached—as in, either a dispassionate ruthless bastard or an ascetic. No airbender has been an ascetic since the genocide. And while Zaheer's not an ascetic (ascetics don't smooch tall firebenders in the shotgun seats of snowmobiles), he is a dispassionate ruthless bastard; the last aspect of his life that he couldn't be dispassionate about was a certain tall firebender who'd just removed herself from the picture.
In parallel, There's also a bit of a darker element to the whole "Enter the Void" concept, that which speaks to death. There's a lot of underlying tones that point to the idea that Zaheer was able to fly because he was able to accept death (falling off the cliff). I'm not sure how accurate it is, but "Void" can also refer to the emptyness that comes after death in some literature. If that is in fact part of flight, then Aang would not be able to access it, because he would not be willing to accept death as the alternative to flight, where-as Zaheer would rather have died than not been able to fly and escape with korra. Floating it out there though, could be a little off basis, but a perspective.
Well first off he DID open the 7th chakra, just not at that moment. Opening the chakras is what allowed the avatar to have control over the avatar state, not just mindlessly rage in it like we see Aang do earlier. At the end of the Crossroads of Destiny after meditating in the crystals (he even says "I'm sorry" to Katara I believe) and again in Sozin's Comet that chakra is open. All of the past avatars achieved this at one point as well. As a few of you nicely pointed out opening the 7th chakra would be part of attaining flight, but definitely not all of it. Just because he let go his earthly tethers he didn't enter the void or empty himself to become wind.
I actually do think they're essentially the same thing, it's just that Aang only has to let go when he wants to start glowing, but Zaheer doesn't activate any special state to use his flight.
In theory, if Aang believed himself to be capable of flight, he should have been able to do it by simply entering the same mindset as he is in during the Avatar State. It sounds lame, perhaps even kind of is, but it's made apparent that a huge difference between Zaheer & previous airbenders is that the latter believed Laghima's flight power to be the embellishment of legend, while the former thought it was fact.
No. I think you need to do more than just open all the chakras to fly. I think you need to free your mind in every way. I think the "enter the void" thing was foreshadowed by Zaheer's speech "if you base your expectations only on what you see, you blind yourself to a new reality", so basically it's kind of a The Matrix route in this XD. So to be able to fly, you need to rid yourself of all doubt and sacrifice all earthly desires and attachments. I also think the Sky Bison have similar way of flying.
In Buddhism, the physical world is considered an illusion, that's why you're supposed to let go of earthly attachments to begin with. Ridding oneself of earthly attachments is what allows you to see the "true reality."