The riser is installed first for the reason that you want a nice tight fit along the top of the riser to the tread above it. There is always the possibility that there is a small variation in the width of the riser boards or the height of the notches cut in the stair jacks. The back edge of the tread can then be slid right up to the riser for a nice tight fit. Since the tread bull nose generally overhangs the next riser down small variations in the width of the riser or the width of the notches in the stair jack are not noticeable.
There are other considerations as well:
- As shown above it should be obvious that the treads can be easily removed if they require repair or replacement.
- If you try to simultaneously get a perfect fit of the riser on both its top and bottom edges you will have a huge task in hand.
- An experienced carpenter will install the risers and treads from the top of the stairs down so that they can always be fitting the next riser against the bottom of the previous tread.
When alternate installation methods are used additional molding is sometimes installed to cover up imperfections of the fit. Some folks may really like the added cove molding but it does detract from the clean lines of the installation as shown above - particularly when a thicker tread is used. Do remember that when ascending the stairs you get a straight on look at the quality of the workmanship. Lastly when the riser sits atop the tread some of the tread depth is lost.
(picture borrowed from this other question)