This is a tough point to explain. On one level, there isn't a difference at all. A person is expected to have faith in whatever they are pledging to, and will honour that pledge. What if the person is not of that faith? Then this pledge will mean nothing to them. Only the morals or codes they adhere to themselves will guide them, not the promise.
But they have a choice. They may swear on the bible (christianity), the koran (islam) or simply the Constitution itself.
I see what youre saying but isnt "god" a general term used for every religions head dude?
They are not required to say that. The actual oath is:
"I do solemnly swear (or affirm) that I will faithfully execute the office of President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States."