New game and restart
New game gives you a different puzzle. Restart puts the same puzzle back to the start. Your marks come off. The clues stay.
Use New game when you want another grid. Use Restart when you want a second run at this one — the same cells, empty again.
If you only want a pause, leave the board and use Continue on Home. Continue keeps the marks you already placed. Restart does not.
A solved puzzle is already finished. Opening a new game does not take that finish away. An unfinished board you still want should be left for Continue, not restarted.