By “head-piece,” I mean the part with the embouchure hole. As far as I know, the reason to clean a flute when you’re done playing is to prevent the keypads from absorbing the moisture. Since the head doesn’t have any keyholes, why do I need to clean it?
Bacterial buildup? Possible corrosion?
Oh…head-piece on THAT kind of a flute.
In that case I got nothing.
Basically what Mr. Blue Sky said. You want to get rid of any moisture and residue from saliva before it has a chance to form buildup in the flute.
So you don’t put a wet, slobbery head joint in with the body of the flute. The moisture is going to be absorbed by something – probably the readily available pads.
'Cause otherwise it’ll get absolutely disgusting, that’s why! 
Hey Speaker, now that the question has been answered, would you care to share with us what you have been up to with a flute – how long you have been at it, what you like to play, etc?
If you’re curious about my reasoning, it’s more than simply laziness. My flute-cleaning rod (something like a giant pipe-cleaner, I suppose) is stored in the body of the flute, so if I clean the head, which has the most moisture, by far, I have to wait for the rod to dry before I put it in the body for storage.
As to my playing, I’m not very good. I began in elementary, played for three years, and then quit in highschool. Five or so years later, I’ve borrowed my girlfriend’s flute and begun playing again. That was almost a month ago, and I’ve got back some of my previous skill. Right now I’m just playing the few-bar-long songs in the teaching books and putzing around with my old music.
Please don’t store your cleaning rod inside the body of your flute. As you’ve noticed, the fuzz on the rod is quite damp. Then, you store it right next to the most moisture sensitive areas of your flute. Better to store the cleaning rod outside the case, say in a zippered overwrap.
The package of the rod says that it should only be stored in the body.
Okay. It (the manufacturer) is probably thinking that the rod will absorb the moisture rather than the pads. Who am I to argue? Many of my students do the same thing. It’s just that serious flute players I have known (and sax players, who have a similar thing) say not to do that. Honestly, it may not make that much difference anyway.
As a music educator, I am glad to hear that you have picked up the flute again. It is truly a lifetime experience, not just something to do in school.
I didn’t mean to be snippy. I think the manufacturer’s intent is for you to dry off the rod (which I do on my pants) and then store it in the body so that it absorbs any moisture left on the pads.
I really enjoy playing, but I’m at a bit of a threshold. My rhythmic skills are holding me back from playing anything more complex than, say, the Star Spangled Banner.
I used to play sax/clairnet. If I didn’t clean the mouthpiece regularly the mouthpiece got disgusting. We called it “Gunge”. You wouldn’t want to put it in your mouth. 
Hey, I teach middle school band. You don’t have to apologize for being snippy. A friend of mine likes to say, “I don’t have feelings – I teach middle school.”
Go to a music store and buy the book Essential Elements 2000, published by Hal Leonard. This book has two volumes. Obviously, book 1 is the beginner level. You may like Book 2 better. The real reason to buy this book is that it comes with a demonstration copy of SmartMusic. This is a wonderful practice tool. It includes a tuner, rhythm charts, etc. You play an exercise in the book into your computer and it evaluates your performance. It’s really cool. It’s meant as a classroom method, but I don’t see why an intelligent adult couldn’t use it to improve reading skills. The book is cheap – under $10. It’s a great way to get to this SmartMusic program. Check out the linked website for more info on SmartMusic.
Good luck and let me know if I can help.
Every teacher I’ve had has told me not to do that. (I was a pretty serious flutist throughout high school and college. I mostly put it aside because it seriously exacerbated my TMJ issues. Oh, now I’m sad. I just got a nightguard for my jaw, though, maybe I can start playing again!) I always used the metal rod with an old napkin through the eye to clean out my flute. Turn the napkin over the head of the rod so that you don’t scrape the inside of the flute. I use a fuzzy thing for swabbing out my piccolo, but only because the itty bitty rod it came with is too short for the body. But I store the pipe cleaner thing seperately.
BTW, in case you didn’t know, the best thing for unsticking a key is rolling paper (not the sticky side, though, of course!). Never use fabric or thicker paper, you’ll shred your pads.
Anyway, yes, clean out the headpiece. Otherwise it’ll get all nasty and bacteria-y. Also, make sure you rinse your mouth out or brush your teeth before playing.
Three cheers for Kyla, backing up my post. Half a lifetime in band hasn’t been for naught.
Hip hip hooray!!
But it sure gets a lot of doubious looks when you open your case in high school and there’s a pack of rolling paper. “Dude, you play flute and you’re a stoner? Double loser.”
-rainy