When we were shopping around for a minivan in late '11, my wife liked the Sienna's exterior styling over the Odyssey's. I like to keep an open mind so we checked out both models in detail. The second we sat in the Odyssey we both noticed how much nicer the interior fit and finish was compared to the Sienna. Yes, the "lightning bolt" on the Odyssey's belt line is a bit different, but honestly, we knew that we'd be spending more time sitting inside the minivan than staring at it while sitting at a cafe across the street.
We've got 3 kids ages 7, 5, and 2. At one point we had two Britax Marathon seats (they're chunky!) side-by-side with a booster seat in an outboard position all in the middle row. Lately we've been volunteering to drive for our kids' school field trips and the additional 2 positions at the rear bench with LATCH points is nice to have. I don't have the exact figures, but for every Sienna in our kids' school there are probably 4-5 Odysseys.
Another pro for me is the better gas mileage that the Ody gets; we get about 20 mpg mostly in the city while dropping off the kids and running errands. Once we hop on the freeway, though, it's normal for us to get 26+ mpg average for the entire trip. I don't accelerate like a grandma and I usually drive 70-75 mph. Our friends' Sienna only gets 20-21 mpg for the same trip on the freeway.
I've come to realize that my biggest gripes about the Ody are because of my sport-oriented cars. I hate how slow the steering ratio is. If you start pushing the van the steering wheel gets numb and squirms a little at anything past 8/10. I'm annoyed at how the Honda engineers listened to the focus groups and softened the suspension damping, especially in the rear. We keep the rear bench collapsed and only keep a 30-lb bin full of miscellaneous crap for the kids in the trunk, just behind the 2nd row. My wife and I are of average weight and our kids don't weigh much. I can feel the rear end hitting the bump stops a bit too frequently for my taste. Short of getting custom shocks/struts, the only band aid solution that I plan on doing is installing a set of bags in the rear.
We only had to bring in our Ody to the dealer one unscheduled time to get a TSB performed to reprogram the transmission shift schedule. It was shifting a bit rough (almost like someone let the clutch out carelessly during an upshift with a stickshift) and there was a coarse vibration during some 2-3 and 3-4 upshifts at > 2/3 throttle input. They fixed it in a few hours and that's pretty much the only problem we've had.