A manual transmission with a "conventional" clutch, not a multi-plate version, will have a smooth metal surface on the flywheel and a smooth metal surface on the pressure plate.
Friction material similar to that of brake pads is bonded to the clutch disk, which engages the transmission.
When the clutch pedal is depressed fully, the flywheel/pressure plate is free to spin with the engine speed without transferring movement to the transmission. As the pedal is slowly released, the pressure plate closes on the clutch disk, providing force to be applied to all three components.
It's important to note that using the clutch to hold position is akin to pressing on your brake pedal while you drive, creating excessive wear and extreme heat. You will eventually burn out the clutch, preventing the automobile from moving, and destroying all three components.
The web site How a Car Works has a few useful images and a full explanation, photos below from linked site:
This photo shows the clutch with the pedal depressed fully, while the one below shows the pedal fully released:
Your question about why one specific action creates movement on the roadway while the other does not is likely that the clutch position has changed from one to the other without being detected, as each action should create additional movement. You may not be able to "maintain the sweet spot" when you increase the throttle. Also, the additional heat from running a slipping clutch may also be destroying the clutch lining preventing movement.
If you wish to avoid expensive repairs and have a working vehicle for an extended period, cease to slip the clutch in the manner you describe. Brakes are to prevent slipping back, not clutches.