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!