First, let's define both BAI and BMI. Body Adiposity Index (BAI) is a new testing method that is still in the process of being tweaked. It determines your body fat level, without stepping on a scale, by looking at your hip circumference and your height, your body weight is not used in the formula. Body Mass Index (BMI) has been around since the 1840's and is used by physicians and researchers to also determine how much body fat someone is carrying. A score of <25 kg/m2 is considered healthy while a score >30 kg/m2 is unhealthy. BMI calculates the ratio between your height and weight but the measurment is not always accurate. It cannot be used in certain populations like individuals who are obese or who carry a high percentage of lean tissue (muscle mass). A great way of getting a picture of your overall health is to use BMI with a % body fat measurement and your waist-to-hip ratio, notice that body weight was not even mentioned.
Here is an abstract in the Obesity Journal on BAI.