Sit yourself down
Eventually, I am wondering if you as a native speaker feel any difference between those or not.
There is no difference in literal meaning. Sit down is the more common phrasing. You would use Sit yourself down only under certain circumstances:
1) Regional accent. This formation is characteristic of certain regional accents or dialects.
2) Informality. The use of “yourself” would rarely be used in a formal setting; it might be used to convey informality: “Why don’t you sit yourself down and stay a spell.”
3) Emphasis. It can also be used to add emphasis to a command. “Bobby, you sit yourself down right now and stop that foolish talk.”