.380 generally refers to a cartridge used in a .380 semi automatic pistol. The cartridge is also referred to as 9x17 or 9mm Kurt (short). Common guns that use this cartridge are the Walther PPK, Walther PK380, KelTec .380, Ruger LCP, Sig P230/232, Beretta 84, Bersa .380, etc.
.38 is usually referring to the .38 special revolver cartridge. Think snubbies or Smith & Wesson model 10.
Now is the question to compare these two cartridges? .380 is commonly loaded with a 90 - 102 grain bullet. .38 Special is commonly 110 - 158 grain. Depending on the barrel length the .380 in an auto will provide more velocity. I personally consider the .38 special in a snub or .380 in a medium sized automatic to be the minimum for a self defense handgun.