First off, I think the 3DS XL is by far the superior model for Nintendo's system and it's what I'm chosing to base this comparison on. Secondly, I have not played hardly any Vita games so, just to be clear, this is my impressions of my hardware and system software.
From a design standpoint, the Vita looks a little more "adult" but I think they are both about equally refined in most elements. My largest complaint as far as appearance about either is that the Vita's screen is not centered vertically on the device which just looks a little funny to me but does not affect gameplay. I am also not all that keen on the 3DS XL's two-tone color scheme but it does look much better in person. As far as practical concerns, the volume slider on the 3DS XL drives me batty because I am constantly moving it accidentally (why not put in on the bottom like on a DS Lite, Nintendo?) but I do prefer being able to suspend the system simply by snapping it shut whereas on the Vita you have to reach around to a rather small power button on top.
Screenwise, the Vitas screen is really incredible. It is large, very sharp w/ no scanlines and perfect backlighting. It is bright and fantastic. That said, I feel like the 3DS XL is more impressive. The resolution isn't nearly as sharp--and it's not as capable graphically--but that second big screen really provides. That is w/o taking into account the 3D effect on the 3DS XL, which to me does add quite a bit to the experience. The Vita feels like it has a big screen for a handheld but the 3DS XL truly feels huge to me.
Controlls are another big difference. I think the Vita has the advantage here, at least theoretically. Both systems feature d-pads, four main face buttons, shoulder buttons and touch screens but the Vita has dual analog sticks and an additional touch pad on the back. The reason I say theoretically is because the face buttons on the Vita are downright tiny, which makes them somewhat awkward, and the analog sticks don't quite feel as good as the slide pad on the 3DS XL. I'd wager most fans of modern FPS and action games will prefer the Vita for the dual analog sticks but I personally prefer the way the 3DS feels in my hands.
As far as the included software, I have to give the nod to the Vita here unequivically. Navigating and viewing on the same screen just feels a lot more natural to having a touch screen on the bottom and another screen for input (though this setup works great for gaming). For starters, the Vita has an actual usable web-browser which alone puts it a class above the 3DS XL but the inclusion of things like Google maps means it might be a practically useful device for that undoubtedly relatively small group of people who have a handheld gaming device and not a smart phone. Additionally, at this point, the Vita has what I feel is a much better selection of games available for download and even though Nintendo is increasing its selection, I have a feeling Sony will have no problem continuing to compete.
The biggest thing against the Vita right now is it's still a relatively new system and does not have the kind of depth to it's gaming library as Nintedo has w/ 3DS. For this reason--and since I feel they are on relatively even footing as far as hardware quality--I'd have to recommend people who have neither to start w/ a 3DS XL. There are a lot of people for whom dual analogs are completely necessary and for those people, I might recommend and waiting for that killer FPS that will undoubtedly come out of Vita. Otherwise, the 3DS XL is going to scratch your handheld gaming itch and it can scratch it right now.