It is now required to have a grounding conductor run back to the main panel from the subpanel. The ground and neutral are kept separate in the subpanel. This is to ensure that the circuit breakers will trip if there is a short circuit to ground. The resistance in the earth is too high to carry enough current back to the main panel to trip the breakers.

You also need a ground rod connected to the grounding terminal strip in the subpanel for lightning protection.

Many many years ago it was acceptable to just have the ground rod without a grounding conductor back to the main panel. That is no longer permitted in the National Electrical Code.

Remove that green screw in the upper right part of the neutral terminal strip