Superman or Batman without a cape smacks of heresy. I shudder to think the powers that be almost decided to remove Supe's cape for his latest movie flight.
The consensus is Superman started the cape fad. The golden age Superman's costume was inspired by a) circus strongmen and b) acrobats. Historically some acrobats used their capes in their acts to control their falls. This makes sense if you remember Superman originally didn't fly but make extraordinary leaps. So I suppose any human grasshoppers could use a cape likewise.
The movie The Incredibles gave capes a bad reputation listing numerous wearers who suffered due to getting their cape snagged. I think they have a point however a super who can't tear his cape off in a pinch or design a quick release for it isn't much of a hero.
Capes can keep you warm on a cold stakeout or protect you from the elements. I suppose a world with flying people also has room for bulletproof capes to add to a hero's protection as well. Fireproof fabrics are another option. Even if your hero is nigh invulnerable that person he's rescuing from a burning house isn't. Wrapping them in a cape might be a good idea.
In days of swordplay many fencers used their capes to fake out their opponents and I suppose a full length cape could draw an attacker's fire away from your body. You can also snap it in their eyes if you like that sort of thing.
A hood is a popular accessory to a cape. On the plus side it obscures your features without the need for a full face mask and helmet. The drawback is that it interferes with your peripheral vision. Having a thug come up out of your blind spot and clock you won't make you seem terribly mysterious.
Some supers have secret pockets in their capes to hold items. I wouldn't since I advocate quick release capes. It's no fun leaving the her holding your cape when you make your escape if your wallet is in it (or even your house keys.)