You can get something like 1% more HP for every 10 degrees F reduction. The flip side is that the longer intake tract will be more restrictive. For any properly built intake, the temperature factor is dominant. There are intakes available that seal to the hood (although how well probably varies a lot), so you can see an open filter with the hood open, but it still acts as a CAI. Depending on the airflow in the car, sometimes the underhood air temp is "not that bad" at highway speeds (where you need HP), but it's generally terrible sitting at idle (where you don't make any HP).
The aftermarket intakes aren't done solely for appearance. Some people prefer the sound, compared to an OEM intake system that is designed to quiet the engine (especially if you have a turbo or supercharger). There are cases where the OE intake is restrictive enough that swapping it out increases power, but generally only if you can draw cold air. There are exceptions to every rule, of course. Sometimes the engine swap or whatever mods means there is no factory intake that will fit.
Sometimes the aftermarket filters are dangerously poor at filtration, so do your research. Paper is fine as filter media, you just need a lot of surface area. Many an engine has been saved by a paper air filter when the intake starts sucking up water.