Hi,
I got both the Duet and the Pico each about a week or so into their being shipped. I got a chance to get familiar with them. The two DAC sections are pretty different. The Apogee is more transparent and to my ears is ahead of the Pico. For whatever reputation and history is worth, Apogee has been advancing their conversion technology for a long time now to be something of an industry standard.
The Duet's amp section on the headphone out is a slightly different story. It is truly excellent and a significant step up from the "Mini" line which I also owned. However, I think most people here will prefer an offering by RSA, Xin, Meier, etc. It is not glamorous, it is neutral to the point of being a little tasteless. You understand it's designed to be. The headphone out is designed for recording musicians to hear their mix accurately. You want to give them water not wine. If you flatter them with it, when it goes to print and is played elsewhere they won't be happy as they get sober. Headamp designers go for accuracy at the bench, to be sure, but they strive for euphony. They want it to sound true, but it's more important that it sound good. That's what makes them and you happy.
To get the best of both you really have to go line out of the Duet into your amp of choice. The suggestions for 2X1/4" into 1X1/8" work just fine if you are stationary. If this is going to be a transportable as it is for me, the breakout cable is a lot of extra bulk. If you are not a musician, this is great! All of those connections would have been inside making the Duet twice the size. This way, if you don't need them, cut them off and cover the ends. If you can work a soldering iron, re-terminating the lineouts into a single 1/8" plug is a snap and you're all set. (If you are both musician and headamp enthusiast, Apogee sells replacement breakouts for 24.95.) If you want to be completely audiophile about it, contact Apogee tech support and they can send you the pin out schematic as you talk.
Hope this helps.