Both systems are humane compared to the earlier status quo, but an even more humane system could certainly be devised. To answer your question, in the case of decapitation there is reason to believe death is not instant but rather that the severed head may remain aware for a few seconds. This is faster than hanging, but still not instantaneous.
A shotgun slug straight to the skull would be pretty damn close to instantaneous, but that's got a few problems. In addition to the psychological damage inflicted on any witnesses to the execution, it will have more inconsistent results. What happens if the shot is poorly aimed and 'merely' removes somebody's jaw bone? Done correctly, a shotgun slug would provide a more instantaneous death. However it's a less foolproof system. That's just an example though, it's easy to conceive of systems that are simultaneously more instantaneous than a guillotine but are simultaneously are more reliable.
Edit: re inert gas: Yes, though again there is the matter of reliability. An inert atmosphere will kill a human reliably, but the apparatus used may fail to provide an adequately inert atmosphere. This probably wouldn't happen with a mechanism designed specifically for execution, but it does happen a lot in cases of attempted suicide. Inert gas suicides will sometimes be botched, with the victim receiving just enough fresh air to survive the experience (typically with severe brain damage caused by the oxygen deprivation.)
Ultimately I think the whole matter is about risk tolerance. How much risk of executing an innocent man do you tolerate? How much risk of a botched execution do you tolerate? How much risk of releasing a guilty man do you tolerate? These are questions with subjective answers, and for many people their answer will be to tolerate no risk, e.g. abolish the death penalty. Other people are willing to tolerate more risk, so for them the answer might be "bring back execution but modernize the methods". Generally people on one side view the thought processes of people on the other side as alien or obviously defective, but I think that's generally not the case.
There did seem to be some resistance to the idea that executions would be simply drifting off - suffering seemed to be a desired part of it :-(
Edit: For the record, I'm not in favour executing anyone but if it has to be done I don't see why people have to suffer.
Many proponents of the death penalty want to see pain and suffering. They aren't remotely interested in humane execution.
1) How much damage a brain can take before it's incapable of experiencing.
2) How fast a brain can have an experience.
3) How long it would take for a supersonic shotgun slug to reach point 1.
I'm extremely skeptical that point 2 is fast enough to meet the deadline imposed by point 3. I suppose I can't say it's impossible, but it seems very unlikely.
For that matter though, let's suppose point 2 is fast enough to meet the deadline. We don't actually know if that is a more unpleasant death experience than being hanged or decapitated. We can speculate about which experience is the worst, but none of the dead can actually report back to us with real data. My personal suspicion is that decapitation is the most unpleasant, since there is a strong possibility you will be conscious long enough to experience your situation, and that experience will be truly unpleasant. But who knows.
But what if an intact brain is an important part of navigating the first 10 minutes or so of the experience of being dead?
I prefer not to have my brain mushed at all, and let it very slowly decompose. Certainly the first hours of being dead, while it's still possible that electrical or chemical activity in the brain is ongoing, I prefer for it to be intact and at peace.
In my model of the world, perhaps a brain slowly ramping down is a pleasant experience. That's totally conceivable to me. However the brain being instantly disassembled faster than it can have experiences would be the absence of experience, be it a pleasant or unpleasant.
But even then, we can't know the difference in experience for brain matter sprayed all over the room vs contained in one place.
A materialistic model asks us to question what happens when the material in question is rapidly separated and damaged, no?
Are individual pieces of cortex capable of experiencing distress for the minutes or hours in question? We have no idea.