I have the Wii Fit Plus. It will show your BMI when you do the daily check in, but it will not show your weight unless you click on a button after the test. You can also skip this without impacting anything, if you want to do it less often or when no one's around.
I will say I don't think it is a great workout in terms of cardio. Good stretching/yoga, muscle stuff, and it's the games are fun and slightly physical (or in the case of hula hoop or cycling very intense).
But overall I got EA Sports Active More Workouts and it is far superior to work up a sweat. If you are looking to replace the gym for a day (which I wouldn't ever choose to do with a video game) I would do EA Sports Active. If you are looking for something that is fun and gets you going a bit on days you work out instead of vegging out in front of the TV, Wii Fit is better.