the skill of the rider is more important than the bike. I am assuming we are talking typical XC type of bikes. You just can't buy climbing ability.
sometimes it is the HT and sometimes it is the FS bike. It just depends on the terrain. In the early days of FS, people would talk about how much of a dog a FS is in climbing but with the modern suspensions and settings, it's hard to tell the difference. The prime example would be the specialized epic or the FSR elites with the brain rear shock. you can set those babies up to nearly be like HT or the suspensions where you can lock the rear shock or at a minimum have propedal.
Time in the saddle and how i felt after a long FS or HT ride determined which bike i would ride. I can no longer ride a 26" HT without it killing me so I went FS but now I can ride a steel HT 29er pretty much all day too.