Because nobility could chose a fine blade, while peasants were beheaded with a less efficient axe (sometimes requiring two to three chops if one were clumsy or did not possess the required strength), this method was deemed inhumane and unfair. The guillotine was efficient for everyone.
The advance of STD and blood-related diseases would make clean up after a beheading a messy business, indeed. The upside though is that it is economical (requiring no electricity, and other expensive precautions - only sharpening after every execution to ensure a clean cut the first time down). Furthermore, the prisoner need not even be conscious throughout (which would prevent the shocked look of horror on the face, as the brain retains at least enough blood for conscious thought for up to a minute afterwards before dying from blood loss).
(If I've freaked you out, don't fret! I studied this topic for a book of mine, so I know more than the average person does on it).
Beheading is barbaric and unnecessarily disfigures the corpse. If I had to be executed, I would choose a single bullet through the heart, with no blindfold or bag over the head or hands or feet bound. I wouldn't want my last seconds to be in the dark.
Outdoors, summertime, on nice cut grass, warm sunshine on my face...perfect...and not unnecessarily trumatic to any loved ones I might be leaving behind...open casket funeral still perfectly doable.