The people of the OT were under the Abrahamic covenant which included the law which addressed the sabbath and how to treat it.
The people after Christ's death and resurrection are held under a different covenant. This covenant makes the law of Abraham unnecessary. The law of the Jews was not meant to save them; it was meant to show them how imperfect they are an how much they are in need of a Messiah. Christ's covenant was an answer to Abraham's covenant. Therefore, Christians changed the date to celebrate Christ's resurrection and the new covenant.
It does start to get even more confusing ...
... when trying to decide which day is Sunday!
What about the people that were born before Christ but died after him?