Within a year, after its debut, smokeless powders were introduced and used in the .38 Special. The maximum pressure for a standard load is 17,000PSI. Standard +P loading would be around 20,000PSI. Common weights for the .38 Special round will vary from about 110 to 158 Grains. The height of a loaded cartridge is 1.55 inches (39 mm).

The 9mm round was introduced in 1902 by George Luger. Thus the term 9mm Luger. Which is also known as 9mm Parabellum or 9x19. This cartridge was designed to use the more volatile, higher pressure smokeless powder, thus the shorter case length. The maximum pressure for a standard load is 35,000PSI. Standard +P loading would be around 38,500PSI. Common weights for the round will vary from about 115 to 147 Grains. The height of a loaded cartridge is 1.169 inches (29.69 mm).

Now pause a minute, lets stop right here. Both bullets are equal in bullet diameter, the 9mm case is about 3/8 of an inch shorter than the .38 Special but boasts double the pressure.
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While I feel either is sufficient to get the job done, the 9mm boasts a much better advantage. The 9mm has 3x the ammo capacity and produces near double the energy.