I would say water alone is a very selective media. In microbiology, when we wanted to test for certain organisms vs others, we would sometimes use agars or growth media that contained a very sparse or few types of nutrients - a nutrient deficient media.
They more basic and fewer nutrients you include, the fewer types of organisms there are that can live there (always a few though, it seems). The more complete the growth media, the more types of (potentially harmful) of organisms that can grow.
They will rot in water also. If it's breaking down cellulose, it's probably a yeast/fungus. But bacteria and (and everything else) would love the sugars in the sap and they can quickly infiltrate the stem - this is what makes your flowers wilt in plain water.
Florists use preservative solutions that usually contain sugar, acidifiers and biocides to make commercial cut-flowers last longer. The chlorine in the tap water mostly dissipates in about a day once it is dispensed into an open container, but it should help to disinfect the cutting if replaced regularly (every other day at least).
The UV light and oxygen levels are also different in a saturated medium vs water jar, leading to more anaerobic conditions if the medium is too wet (or if you never change the water in the jar).
I wonder if adding a phyto-friendly disinfectant, like a quaternary amine, and perhaps sugar/honey (as suggested once by @"Figaholics"#29 ) would increase rooting percentage in water, along with daily water changes.
The trick part seems up potting, since the adventitous "water roots" breakoff pretty easily and the transition to soil seems tougher than if they were rooted directly and the cutting can remain in the same container (e.g. 4"x9" treepot) until rootbound with significant top growth and still be up potted with minimal root disturbance. This seems like less work and less chance for failure.
The key, as has been mentioned, is getting the initial water content right and keeping humidity very high (in the root zone), temp around 75°F and "gentle" lighting. We hydrate coco coir blocks to field capacity (myco definition, i.e. squeeze test) and add about 30% (by volume) dry material (perlite) for a final water content of around 70% field capacity. I think most growers have a "method" or go by feel (after you kill enough cuttings, you start to get a clue
