In The Legend of Korra, we learn that
Flying unassisted is an incredibly difficult spiritual skill that very few non-Avatar airbenders were ever known to do.
Why is this?
We have previously seen multiple firebenders fly by using their fire to propel themselves.
Is there any reason why airbenders couldn't do the same thing with air instead of fire?
I think you need to make the distinction between flying by either method you mentioned and levitation.
Firebenders fly via directed propulsion; like rockets.
Airbenders fly via manipulating air currents over wings, like planes, or by propulsion (a swirling ball of air is applying force against the surrounding air).
Guru Lahima, and subsequently Zaheer, achieved levitation; flight without propulsion or manipulation of air currents. There isn't an explicit canon reason for what's going on, but the Avatar universe embodies a mish-mash of various religions. Two important contributors are Taoism and Buddhism, and for your question, you would want to take a look at Buddhism.
Not only do the Airbender monks look like their actual Buddhist counterparts (down to how Avatar Aang was chosen and how the Dalai Lama was chosen), but they share many of the same tenets. Specifically, you should look at Śūnyatā, the concept of "emptiness" or "voidness." To mangle a multi-thousand-year-old religious concept down to something more digestible: Reality sucks, and to break through the veil into heaven, you have to let go of reality in favor of meditation to achieve enlightenment.
Zaheer had a lot of time to meditate, and his last tether to reality was his interpersonal relationship with P'Li, the combustion bender. Once he presumed her dead, he achieved a stage of enlightenment that allowed him to "be empty" -- and then just levitate away.
Aang was also instructed by a Guru to "let go" of his earthly desires in order to complete his Avatar training and unclog his final chakra, but he refused to do so. Had he done so, it's possible he might have achieved a similar technique to Zaheer.
TL:DR = The religions the show heavily borrows from have a basis for achieving forms of enlightenment through inhabiting a form of emptiness through meditation. Presumably the enlightenment allows the Airbenders who achieve it to levitate at-will, and Aang might have been capable of something similar had he followed through on his Avatar training with the Guru.
Especially powerful airbenders, those who have mastered the element, can manipulate the air around themselves, which would enable them to fly through it. But there just aren't that many airbenders left. Aang is the only airbender in The Last Airbender and I think there are only Tenzin and his children in The Legend of Korra.
So we just don't see enough airbenders that have really mastered the element enough to fly unassisted.
Firebenders generate their own fire. Firebending is all about overpowering force tempered by the unflinching will to accomplish tasks and desires. So a firebender just needs to keep pumping out the fire with enough force and they could fly all day.
The key to Airbending is following the path of least resistance, so an Airbender would naturally fly with their staff on the air, as they don't generate their own.
Fire benders can't actually 'fly' with their power, that is only possible on the day/night where Sozin's comet comes. In reality they are simply propelling themselves with jets fire and can't sustain it for very long (this is done by Korra and General Iroh of the navy [Zuko's grandson]). Airbender's are much the same, they usually use air to perform jumps that carry them several meters, as seen performed by Aang and Tenzin and most of the other new air benders. Zaheer and most likely Guru Lahima also have given up their tether to the world in some way. This allows them to fly. Some might say Aang could fly, however he actually couldn't. During the fight with Firelord Ozai, he encased himself in an orb of swirling air. This can't be done by all air benders or if so, it can't be sustained for as long as Aang did (remember when Aang 'flew' he was in the Avatar state). Only the Avatar can because he/she has the strength of all previous Avatars within themselves helping them when they enter the Avatar state. Dang when not in the avatar stat has been seen to run out of air scooter. I hope this has been helpful.
