I don't use carefresh, being a Pine guy myself, but I know of many who do. When it gets wet, it stays wet. I think it stays wet longer than Pine does in similar circumstances. Since there is no chance of ammonia conversion in a dry environment I would "guess" that Pine would promote less ammonia.
Of course my guess, is just that.
Now if you keep or allow your Pine to stay wet, then sure, ammonia is inevitable. But in the short term, knowing only my own cages and their keeping intimately enough to comment, I have no worries with Pine.