When a profile is copied to a new one and the Weight field is changed, the change is not saved. Here are the steps to re-create:
1) Copy a profile to a new one. The Weight is the same in the copy as the original.
2) Edit the Settings for the new profile (the copy). Change the value in the Weight field.
3) Save the settings.
4) View the list of profiles on the CiviCRM Profile screen. The weight is the same as it was before the attempted change.