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.
The story of Harvest Moon Back to Nature typically revolves around a young protagonist who moves to a rural town. They inherit a farm and start a new life there. They engage in farming activities like planting crops, raising livestock, and building relationships with the local villagers. It's a story of growth, self - discovery, and the joys of rural living.
Well, they share a lot of common elements like the farming mechanics and the charm of rural settings. Maybe it's because the developers wanted to continue the popular concept under a new name.