I get asked this question a lot and here is my take on it...A plank on your hands takes you further away from the ground and asks your body to find stability on a small surface, the palms of your hands. A plank on your elbows allows for more stability because you now have your forearms as a longer base of support.
That being said, there are other factors which play into level of difficulty. Because a plank engages more than just the core (like the shoulders, chest, glutes, quadriceps, hamstrings), often times, it depends on the individual as to which is more challenging. If there are muscle imbalances within your body (which, by the way, we ALL have) then our body compensates all the time. Bottom line is that either position can be more challenging. My advice? Start in YOUR most challenging position first and then modify when you need to. Even if it is for a few seconds, you will gradually build that time and your strength.