What are it's downfalls exactly?

The necessity to hit twice in the head, which requires no small amount of skill is a huge downfall. If they don't get two dome shots in you, you have more than enough time to either trade kills or best them and run for cover while you wait to recover. From optimal range to aim the hand cannon (w/o the increased range) Thorn can be bested by a fusion rifle or shotgun, whatever you're into. From longer range, scout rifles with high fire rate, Red Death, Vex Mythoclast, and a skilled hand using a sniper, particularly with final round (so as not to take the extra time to line up a bucket shot... though I love using the No Land Beyond for the sheer "-BLAM!- yourself!" message it sends) can all ensure the second shot misses the head or is never taken in the case of snipers. I'm not saying it's easy, but I am saying the Thorn is not perfect, particularly in the hands of your average scrub. Yes I am aware of the stats on kill time and precision kill time for the Thorn compared to other weapons. When you lessen your enemy's odds of hitting you, however, the stats count for much less, once again making your encounter a competition of wit and skill.