As far as I'm concerned, "furry" as a noun refers to either "furry characters", anthropomorphic animals in other words, or "furry fans", ie. people who like anthropomorphic animals.
Many fans don't like the noun "furry" in any sense because many people, fans and non-fans alike, associate it with the more adult, perverted, and freakish aspects of the fandom. However, the furry fandom is such a wide and varied thing that it can easily be considered a broken fandom, with breaks upon breaks. One group of fans can't stand another group who can't stand yet another and so on in a virtual Gordian knot of hates and dislikes. Hence, trying to lump any group of fans into any one notion of adult, freakish perversion is pointless, especially since a good chunk of the fandom don't like any sorts of perversions in the first place. About all the various fans have in common is that they like animals that have been anthropomorphosized to some degree.
There is another reason fans don't like "furry" as a noun. It comes across as too cutesy. Many people (usually not the same many people mentioned at the start of the previous chunk of text) think of kiddy, all-ages stuff when they see things with anthropomorphic animals in it. That makes it hard for more serious writers and authors to be taken seriously when they try to present more sophisticated material (not necessarily adults-only, but anything that's more thought-provoking than the average Saturday morning cartoon).