No, they're not the same. Story of Seasons is a new series that developed from the Harvest Moon franchise but has some distinct features and gameplay differences.
They aren't identical. Harvest Moon was the original, and Story of Seasons is a continuation with its own tweaks and improvements. The worlds, quests, and overall feel can vary between the two.
One reason could be to better represent the evolving nature of the game and its focus on the changing seasons and the various activities associated with them. Maybe the developers wanted a name that conveyed a broader and more inclusive experience for players.