In formal writing it's frowned upon because a conjunction joins ideas. Periods separate ideas. It's therefore a bit of an oxymoron to have a period followed by a conjunction. (Well, coordinating conjunctions. Subordinating conjunctions work a bit differently.) And that, dear readers, is why all those English teachers get apoplectic about it. As a stylistic device in informal or formal writing, it has its place. And like any stylistic device, it's very tiresome when overused.
Unlike the preceding paragraph, where I deliberately threw in a couple to goad the absolutists, in my normal writing I have to watch out for overdoing it. I tend to throw them in without being conscious of the fact. So when I revise, I challenge them whenever I notice them and eliminate them if they don't make the sentence stronger. It's good to know your own weaknesses.