I'm going to go against the crowd and say horses are harder. Mainly because you are adding the rider's own physical limitations and mental issues to those of the horse. Obviously, some of that comes into play with agility as well, but not nearly as much.
My own left arm is not nearly as well connected to my brain as my right arm is. So with agility, I have a really hard time "flipping" Meg from my right side to my left; the left arm doesn't pick her up as well as my right, and I end up turning into her too much and she panics. I've quickly learned to counteract the effect by using a different motion with my right arm, and teaching her the verbal cue.
With the horse, I can't make those changes. Verbal means little to him, and is illegal in the dressage ring. Nothing is going to change the fact that I have a hard time allowing his left hindleg to step under because my arm isn't good, other than improving my arm. I think with dogs you are able to make more adjustments to suit the dog and handler; with horses, a lot less options are open.
Riding my horse around the Saratoga racetrack, there is always a bit of concern in the back of my mind that the speakers are going to blast, one of the baby racehorses is going to bolt past, or the carriages are going to go by, and my horse will have a meltdown. At least I never worry about returning alive from agility trials! My friend asked me why I don't get nervous at trials; it is so much less stressful than horseshows for me!
So, as a long suffering dressage rider and a newer agility competitor, that is my opinion. Take it with one of those big blocks of salt.