There may be a high resistance due to corrosion at the battery terminal on the positive side. The thinner wire, I.e., not the one going to the starter, may not be making a good connection.
Normal battery voltage is 12.7 VDC, during cranking it drops below 10 VDC. Jumping from another car, with it's engine running, will keep the voltage above 14 VDC. This is a solid 40% increase in voltage. It can make all the distance in the world when there is a high resistance where it should not be.
You can test this theory with a volt meter at one of the power points. See if there is a difference during cranking with and without the jumper cables.
If you don't have a volt meter an incandescent 12 VDC bulb will do. There is a marked difference in brightness between 10 VDC AND 14VDC. Halogen headlights won't do, nor will LEDs, they are not sufficiently voltage sensitive.