The M1903 is the standard rifle adopted in 1903, modified in 1905 (new sights and knife bayonet, versus the original rod bayonet) and re-chambered for the .30-06 in 1906 (versus the original .30-03.)
The M1903A3 is the WWII version, made as a supplement to the M1 Garand. it uses many stamped and welded non-functional parts (such as the trigger guard, sling swivels, bands, buttplate and so on) to speed manufacture. Many were made with 2-groove instead of 4-groove barrels. It also has an apereture sight mounted on the receiver bridge instead of the complicated mid-sight of the M1903. The finish is not quite as good.
Having said that, let me point out that the M1903 mid-sight is an abomination. The battlesight is set for 547 yards -- much too far. The notches and apertures on the flip-up sight leaf are too small for anyone with less than perfect vision. For a shooter or hunting rifle, the M1903A3 is the better choice. For a wall-hanger or a target rifle (if you have perfect vision) the M1903 might be a tad better.