I disagree with the above answer. Although there is some room for speculation, "neat and tidy" involves the arrangement of things and "spick and span" involves the condition of things.
When I hear the terms "neat and tidy" I think of products like closet organizers and storage bins. These are containers used to put things in their places and puts things out of view in order for a room not to be considered "cluttered."
On the other hand, when I hear the terms "spick and span," I think of Mr. Clean and Bounty and the numerous other cleaning chemicals and cloths that allow a housewife from the 1950s to procure a spotless home for her husband.
Further, I would like to suggest that both of these word sets declare a kind of 1950s mentality about cleanliness (as in, "next to Godliness") that we are beginning to value less in our society which now smiles upon days such as "Casual Fridays." I would suggest this antiquated delineation is the reason why these words aren't used in the frequency that they were used a half a century ago.
Actually, the two phrases mean pretty much the same thing in today's standard English. Neat and tidy often refers to a room/car/home. I.E. "That place is neat and tidy". Often people use the term spick and span for the same thing.
However, spick and span used to mean something that is new. When I think of spick and span I think of someone's remodeled kitchen or brand new car--in that instance, it's "spick and span", shiny, clean, and fairly unused, whereas neat and tidy can refer to something older that has just been recleaned.