The petition to probate the will indicated the deceased died “on our about” the 6th of December. The person’s death certificate indicated that they died on 3 December. Is the difference significant?
Of the two documents, the death certificate is most likely to be correct, but there are always exceptions. The probate court is most concerned about the fact that the individual is deceased. The date being three days off most likely not germane to the petition to begin the probate process. The incorrect date could simply be an error that was not noticed. Even if it was noticed at some point later, correcting the error was likely not considered to be worth the bother.
If there’s real concern about which document is correct, one could search for obituaries or other contemporary sources that would reference the date of death. A newspaper notice of a death on 5 December would suggest that the death certificate was correct.
And if a child was providing the date of death to the court, it is always possible that the date of burial was confused with the date of death. That’s easy for someone to do as well.