Throughout the movie, we're given different versions of the past. I am also not sure if it's ever made very clear why Bane wears his mask.
So, why did Bane have to wear his mask?
From what I've gathered, he has to wear that mask in order to tolerate the constant pain he is in after the mauling he took while helping Talia escape. It feeds a nerve numbing gas to him. That said - it is not really shown in use (gas canister-wise) during the film, and a bit ambiguous. Also seems a bit of a weak spot for such a tough guy.
I've found a quote from GQ magazine:
Nolan has revealed that Hardy's audio-muffling mask is actually pumping Bane's body full of anaesthetic. "Bane is ravaged by pain from a trauma suffered long ago," he explains.
If you'll recall, Tom Conti's character (the old man in the prison) recounts the story of 'the child' who escaped, and this is then extrapolated upon during the big reveal at the end by Talia, who goes on to explain how Bane was her protector and how he was set upon by the other inmates when she escaped. The doctor with the gammy eye tried to help him but his injuries were so severe that he was in constant pain from that moment on.
Actually, I think another good question might be: where did he get his mask from?
The reason that Bane wears the mask is because he has to be dosed with a supply of a drug called "venom". This drug is why he is so fast and so strong. He was already incredibly smart before the drug. The drug was forced upon him back at the prison where the guards and that doctor tested it on him. They did this because they were testing a lot of the inmates and Bane could tolerate it.
The venom is actually supposed to go into his brain via tubes, but that would have involved almost a whole extra movie. In the end the movie had many clashes with the comics but was an awesome movie....this is what Bane is really supposed to receive but after reading the other posts Nolan could've have just rewrote some stuff about pain.
The movie explanation is that the mask provides a steady stream of anesthetic (morphine) to numb the pain from the injuries he received in the Pit prison after helping a young Thalia escape. It seems the same Doctor who was addicted to morphine in the prison tried to repair the injuries to pain unsuccessfully which left him in constant pain.
It's assumed the mask was devised and applied after Bain was rescued by Ra's Al Ghul and the League of Shadows from the pit. They took him in and trained him as a favor for helping a young Thalia escape, it seems all the other prisoners were killed or left in the pit.
Back to the mask, and here is where things get a tricky. We know Bane was beat to a pulp in the Pit prison and the doctor (who was also a prisoner did his best to heal his injured unsuccessfully). It's explained that the botched surgery or treatment of Bane's wounds left him in constant pain. Now medically I'm not sure what type of beating would leave you in constant pain outside of permanent nerve damage or perhaps a cerebral injury that somehow affected pain receptors in the body causing constant pain. Typically someone who takes a severe facial/cranial beating will heal or die from the injuries, or be left mentally handicapped. The type of injury that would leave Bain in constant pain is never fully explained likely due to the fact that such a circumstance is scientifically unrealistic or some specific it would take a long drawn out explanation.
Now, the advantages of the mask. Based on what we know Bane is receiving a steady dose of morphine through the mask, the big advantage here is that (provided he has mastered his motor skills under the influence of morphine, which it seems he has) when he fights the blows/punches/kicks etc. would likely have little effect on him as the pain would be incredibly minor due to the effects of the morphine. There are disadvantages to this however as he could break a limb, rib, or any bone and not realize it which could lead to later complications. Additionally, while blows to the face/head may not cause pain, they will have the same effect as they would on anyone else in regards to loss of coordination, seeing stars. Loss of balance, etc. just because your pain tolerance is nearly non-existent doesn't mean the brain can't be damaged/concussed and make you as vulnerable as anyone else.
In summary, the injury or injuries that would cause constant pain and require constant anesthetic are pretty unrealistic, especially (in the case of Bane) facial injuries. Reality aside, the advantage of the constant supply of anesthetic is that a trained fighter like pain can take a lot more punishment without being hampered by the bodies reaction to pain, he can also strike his opponent more often without being hampered by the standard pain one feels when punching another person bare fisted. While those are the advantages, any hits to his head/cranial area would have the same effect as it would on any other person. Had batman landed a clean hook or a few clean/heavy blows to Bane's head he could have knocked him out.
All that being said I fee Nolan kind of phoned it in with the mask and the explanation. If anything it's imposing but outside of the advantage of being able to take more blows to the body without the pain effect that comes with it, there is no other advantage to the mask.
In other versions of Batman Bane gets his powers from a drug which he is constantly infused with via a system of tubes and syringes which are strapped to him which maintain both is physical strength and aggression with increased dosage having increased effect, at least in the short term.
The Christopher Nolan interpretation has tended to go for a more 'realistic' and less cartoonish feel for the world so his films have virtually nothing in the way of 'superpowers' for the characters and he attempts to create more of a sci-fi than fantasy feel. For example the batsuit and batmobile are based on reasonably plausible military research. Obviously realism in Batman is relative.
So we can speculate that the venom-enhanced bane didn't quite fit with Nolan's vision for the world but he still had some compelling reason to use the character.