Why is the total count of people on a plane given as the number of 'souls' on board?
The primary reason is probably that it ensures there is no confusion between passengers, crew, or infants. Technically, "passengers" is the number of seats occupied, "crew" is both the pilots and flight attendants on duty. So any small children brought on as "lap children" will not be included in the "passengers" count, but should be included in the total number of people on board.

I found another interesting point over on the English Stack Exchange site, which is that dead bodies are sometimes transported as well. In this case, some might consider these "people" as well. Also, in an incident, the bodies should not be confused with the regular passengers.

So, "souls" effectively communicates the number of living humans on board.

There may certainly be holdovers from the maritime influences on aviation as well.