We dined both places in December 2008. Both are wonderful.
From my experience, the breakfast food is better at 1900 Park Fare. The character experience is a little quieter at 1900 Park Fare, too.
If your family would enjoy "napkin waving" and cast members building up the excitement, then they might enjoy the character interaction at Chef Mickey's more.
1900 Park Fare had Pooh, Tigger, Alice and the Mad Hatter when we were there. Of course, Chef Mickey's had Mickey and Minnie.
Chef Mickey's has all guests take a picture together before the meal. While you are eating the photographs are brought to your table, and you can choose to purchase a photo package. I bought key chains and refrigerator magnets, and if memory serves, it was all ready for me to pick up by the time we left.
All in all, I like the food a little better at 1900 Park Fare's breakfast. The atmosphere there is somewhat elegant, because of the hotel. But I like the traditional characters at Chef Mickey's and the tradition of being in the Contemporary Resort (one of Walt's original resorts) slightly more.
For us, parking was easy at both resorts, as long as we had a dining reservation.