510 batteries are regulated to 3.3 - 3.2 volts.
808 batteries are either not regulated or are 3.6ish volts. More power out of a kr808d-1 model battery than a 510 battery and kr808D-1 tend to have more robust automatics and more battery life.
Other than the slim battery difference, the 808 to 510 is almost no different, just how the threading is.