The two expressions mean the same, and for a lot of situations they are interchangeable. They usually differentiate themselves in usage.
Take for example," I am busy at the moment/ I am busy at present". The first sounds more natural, the second not so.
Another example:" This is the problem at present/ this is the problem at the moment." Both are okay, but maybe the first has a sense of urgency to it.
For all intents and purposes, there is no real difference.