We also just went through the Sienna vs. Odyssey debate, and settled for the Sienna.
We had a 2003 Mazda MPV and had so many problems with it, I can't even tell you. We actually did file a lemon claim and got some money back from Mazda. But anyways, without going into it, I'll never own a Mazda again.
We ended up choosing the Sienna for cost. We were looking at the low-end Sienna, the mid-range Sienna, and the low-end Odyssey. My feeling was that the mid-range Sienna compared more evenly with the low-end Odyssey, at which point, the mid-range Sienna was very slightly more expensive than the Odyssey but had more features that I liked. So then, it became a question of low-end or mid-range Sienna. Same engine, exact same safety features (except that the low-end Sienna didn't have traction control). We decided that while the extras in the mid-range model were nice, we didn't need them for the extra $3-4000. So, we settled for the low-end Sienna, which we paid $21,200 for about three or four months ago. So far I'm loving it and have had NO problems. It doesn't have power doors, but I really don't care about that at all. The only feature that I wanted that it didn't have were tinted windows, but I took them somewhere and had them tinted for $185. It has front/rear AC, cruise control, power windows/locks . . . We're not car people. We want something safe and reliable -- the extras just aren't worth the money to us.
So, that was our decision! Having said all of this, it's true that they drive TOTALLY differently. Toyotas have a "cruising" feel to them. They're sort of loose feeling. The Honda feels more like "performance" vehicle. You can really feel the weight of the car -- very solid. I talked to a lot of people, and I don't think it's a matter of better or worse at all. . . Just different suspensions.
Good luck. I'm sure you'll be happy with either car!
It seats 8 (most seat 7). It has stow and go in the 3rd row with a split seat (most have a bench seat) The seats are nice and plush and both side doors open with the remote (a must with 3 kids!). Most of all- it has ever safety feature you can imagine. We looked at every minivan made when we bought ours and this met our needs the best. I am so glad we got it. 
Actually in DH's professional opinion, both the Sienna and Odessey are great choices. They both have about every imaginable feature and hold their value. He sees people trying to trade in other vans and really getting much less in trade than they hoped. Or, they have a hard time selling their vans independently. The Honda and Toyota give you more for your money in resale.
