It sounds like you did not leave the charger connected to the battery long it enough for it to store up enough juice to crank over.
When I jump start a car with another vehicle with jumper cables, I usually let the dead battery charge about 1-2 minutes before I try starting it.
When I use a Snap On jumper box here at my shop, I typically let it charge for at least 30 seconds-1 minute before I try starting it.
Tow trucks are typically equipped with larger and stronger jumper boxes, that is why they were able to jump it so easily.
Considering how small this portable charger is, I would definitely suggest letting it charge a depleted battery at least 1-2 minutes so the dead battery can get enough juice and build up enough cranking amps to turn over.