have you tried re-adjusting the snare wires?  sounds like they're a bit off kilter.  have you thoroughly inspected the wires to see if there's any that are broken?

MrDrums
I've got a Pearl Free Floating 3.5x14 brass snare myself. What I do is get the drum tuned to where I want it and then use very small bits of Moon Gel to muffle in 'problem spots.'
There are going to be certain areas on the drum batter that play significant parts in a sympathetic buzz issue. The cool thing about Moon Gel is that you break off very small pieces that stick to the head and then can move them around easily. That's a big advantage to using MG for muffling. You could also try the very-small-rolled-up-donut of gaffer/duct tape method (do a DC search for 'muffling' and you'll see some good posts). The idea is to be surgical as opposed to general in muffling. Pinpoint a small area (usually near the rim) to apply muffling and that should help.
Keep in mind that I have a Remo Ambassador (soon to be an Aquarian Texture Coated) on my snare which is a thinner head than the Powerstroke. Muffling has a more significant result on thinner batter heads but since you are wrestling with sympathetic buzz it is a  resonance related issue.

Hope this helps

Jim

Loosen everything and start from scratch.

Start tuning the snare head (bottom) and get it where you think you want it. Generally, for the sound you are talking about, it should be fairly tight ... 'firm' ... but not too tight. If you push on the center of the head, it should not have much give ... but should not be "rock hard".

Next tune the batter head (top) so that it's tuned 'medium firm', not as tight as the snare head. If you know your intervals, try tuning the batter head so that it is a Major 2^nd or Whole Step lower than the snare head. If you need more "body" in your sound, increase the interval ... which means you may have to tighten or loosen the snare head, not just the batter head.

Another thing to keep in my is that the snare wires respond directly to the tension of your snare head. Not only is the tension of the snare wires important, but so is the tension of the snare head and how the wires respond to it. This is ESPECIALLY true when it comes to Piccolo snare drums.

One final thought ... you are not going to get tons of "body" from a Piccolo snare. I'm not saying you won't get any ... but not like a drum with more air moving inside it ... and a larger shell depth.

Hope this gives you a place to start.

The only tips I can give is try maybe some different heads,and usually a " buzz " sound will occur when a bottom/snare head is tuned too tight.The drums in essence wants to resonate but gets choked up hense the buzzing sound.This will also occur if your snare wires are as tight as they can possible go and you have to literally force your strainer up.Also once you tune your heads to a nice pop work with the strainer.Loosin it,tightin it.Sometimes how the snares are placed ( tight,loose,or in between ) can have a huge effect n the sound of your snare drum.

Yes, ARCHxANGEL is correct in that the tension of the strainer is an extremely important variable in that 'buzz' sound you are getting (of course, in addition to the tension of the reso head).  

Another way to test if you've got your strainer too tight, is to tap VERY softly with a stick or your finger in the center of the head.  What you are doing is checking for snare strainer response.  If you get a good response from the edge (i.e. rimshot) but your response decreases the closer to the center of the head you get, you may want to losen your strainer a bit at this point (if you are looking for more response in say, ghost notes).  The decreased response generally means you're reso head is choked.  In my experience, this is mainly caused from the tension of the strainer being too tight, in which it is 'crammed' into the reso head thus, choking it.  

I've had the same problem with my Tama/Camco steel snare (admittedly, a larger size than yours, at 6.5X14").  I tried a bunch of different tunings, different heads, different snares, different snare tensions, etc., and I always got the same thing - I'd hit in the center, and it would sound nice and crisp.  I'd hit halfway or more towards the edge, and while the initial sound was still nice, this after-ring would come in, making the snares do that extra little half-second buzz a moment after the initial hit.

I ended up "punting," and put the original internal tone control back on the drum.  It kills me to do that - I don't have tone controls on any other drums of mine, and I don't like the mechanics of the tone control pressing up on the head, in the opposite direction the head moves when I strike it.  But, applying just a little tension to the head with the tone control has done the trick.  I still get a nice, full, bright sound out of the drum, but that after-ring has been eliminated.

I'd try an "o-ring" of some sort - either a manufactured one, or one you cut yourself from an old discarded drum head.  Or, invest in an external tone control, the kind that clamps onto the counterhoop.  

Or, for a really low-budget alternative (but one I like the results of even better), fold up a piece of tissue paper or paper towel, and tape it to the hoop (NOT THE HEAD) so that it rests freely on the drum surface. When you strike the drum, the vibrations will push the muffler off of the head just enough to keep the natural vibrations there, but when it comes back down to rest on the head, it'll cut down (or eliminate) the after-ring you're getting.  It doesn't change the overall sound of the drum quite as drastically as an o-ring.

As always, YMMV.