A filter that lets more air through will offer less resistance to the airflow, so there's less struggle to get air into the engine - hence better filters do make a difference.
When the turbocharger compresses the incoming air, the air heats up. As anything gets hotter it expands (the individual molecules are more highly energised, and thus take up more space). With fewer molecules of air going into the combustion chamber, there's less combustion taking place - less power.
So, if you can cool down the incoming compressed air stream, you'll get more air molecules in, better , more aggressive combustion and more ponies out.
Just makin' sure it's all clear so those that don't know, know, and those that do know, know that I know, and those that didn't know before, know that those that know know that I know that they know.
I'm clear.