What is the difference between a 38 special bullet and a standard 38?

There isn't a difference. Mainly because .38 Special IS the standard .38.

The "Special" in .38 Special is simply a marketing Word. The term was used to distinguish it from the earlier .38 Colt, which is now obsolete.

In modern terminology a ".38" is an abbreviation of .38 Special.

Now, just in case you did not ask the question you meant to...

Another common question is the difference between .38 Special and .357 Magnum. The .357 Magnum was designed as an "enhanced" version of the .38 Special allowing more powder to be stored in the casing and thus deliver more power to propel the bullet. The .357 Magnum is much more powerful and has much higher gas pressures, and thus it is not safe to shoot a .357 Magnum out of a gun designed for .38 Special. Fortunately most modern .38 Special revolvers are designed so that the longer .357 Magnum will not fit in the cylinder. The reverse however is permissible as .38 Special can be fired in a gun intended for .357 Magnum because the magnum-gun is built much more sturdily than required to shoot a .38 Special.