For me 'moderately well' ranks a little less than 'fairly well' for a few reasons. If I say my studies are going fairly well, I'm being somewhat positive and think my studies are going *almost* well. I often throw in the qualifier 'fairly' to be modest, or because I think there's always room for improvement. If I say my studies are going moderately well, I'm being a little more cautious than if I had said I was doing fairly well. I'm also being slightly more self-critical and implying there's more room for me to improve in my studies than if I had used 'fairly well'. I also hear 'fairly well' much more often than 'moderately well'. The 2nd just sounds a far more guarded/cautious statement.

Yes they are very similar. I can't think of a difference.