Problem replacing Slater Pilot Switch with Leviton
The new switch is just the same as the old switch except that the connection between the switch's load terminal and the hot side of the pilot light is internal to the new switch. This internal connection replaces the split done with the wirenut.
Here's how your existing switch is wired:

The single black wire is the Hot wire. The white wire is the Neutral. The split black wire goes to the Load and also supplies power to the pilot light.
New here's a diagram of the new switch:

As you can see, it's exactly the same except the extra connection is internal to the body of the switch.
Let’s label the wires coming to the switch. The black that is not split is Hot, the black that is split is Load and the white wire is Neutral.
Remove the old switch and the split on the Load.
Connect the Hot to the black screw.
Connect the Load to the brass screw.
Connect the Neutral to the silver screw.
Of course, also connect the ground to the green screw.
Done.
Note: the configuration of the new switch is shown by my previous mistake that had him connect the Hot to the brass screw. He reported that the pilot was on regardless of the switch position. With the mistake, the light was directly between Hot and Neutral. I am confident that this answer is now completely correct.