The Model 1903 A1 is the original version of the Springfield Model 1903 with the leaf-type barrel mounted rear sight, but uses a pistol grip stock with a longer length of pull and a higher comb than the original straight grip stock.
The pistol grip A1 stock is much more suitable for use with a scope, and it also gives a better cheek weld than the older stock design even with iron sights. It is basically a target stock design, although it was used for general service as well.
The 1903-A3 is the final service version of the Springfield, using many stamped parts and a receiver mounted aperture sight. They can be seen with the short straight grip stock and the "scant" stock, which has a similar length of pull and high comb to the A-1, but has a less pronounced pistol grip.
Below are my late production 1903 with a scant stock...the A-1 would be similar but would have a more graceful pistol grip, and, below, my 1903-A3 with common straight stock.
The 1903 A-1 would be better for use with a Leatherwood or Unertl target scope mounted on barrel blocks. The 1903-A3 with receiver mounted Weaver scope was called the 1903-A4 and was a dedicated sniper rifle in WWII.