No difference, a long time ago you could do it either way. Same for white moving first, IIRC they used to flip a coin or something to decide who moved first.
But over time, the most popular conventions became rule.
Importance of board setup?
Sort:
If you play with the board "wrong way around" as you put it (with a black square to the right), just make sure the king and queen remain in their correct files. It would no longer be "Queen on her own color", but instead "King on his own color" if the board is wrong way around.
The other problem you'll potentially run into is that when reading/analyzing/playing positions from a book, the "Black Bishop" will be on a white square, and vice-versa.
Best to just set the board up correctly though... and avoid the above issues.
Why is it so important for the board to be set up with white on the right? I've been always doing it to prevent confusion when playing and obviously thats the proper way to do it but whats the difference if you play with the board the wrong way around?