You don't mention ages of your kiddos but if they are young, as in carseats, strollers, etc, then I'd opt for the Odyssey. Not having to worry about them door-dinging the car next to you and your being able to step inside the van while you buckle them in are considerations. Also lifting the stroller into a lower profile vehicle is easier.
I've owned a Pilot for 7 years and if something happened to it, I'd go buy another in a heartbeat. Prior to the Pilot, I drove a Nissan Quest van. One thing we discovered (after we bought the Pilot) was that my daughter got carsick on curvy roads. We had driven those same roads in the van and never had a problem- I guess because the van was lower to the ground, you didn't feel the sway as much. The van also got better mileage and could hold more cargo. We solved that by buying a "roof poof" bag that we use for camping trips when we need more space.
Ultimately, you will be happy with either choice. Have a good trip in your new car!