Continue a puzzle you left open
Continue appears on Home only if you left a board unfinished. It is a way back to that exact puzzle, with your marks still on it.
If you closed the tab, or you went to read a how-to, the board is still there. Open Continue and pick up the same cells.
There is no Continue if every board you opened is finished, or if you have not left one open. The control is not something you turn on. It is a door that exists when you need it.
Use it when you want to leave a hard cell for later. Use New game when you want a different puzzle. The two are not the same.