Bias ply tires have a round profile and high sidewalls, and tend to be for older cruisers. Radial tires have a square profile with shorter sidewalls. A radial tire sidewall flexes more, giving it a larger contact patch with the road when leaned over, and allowing higher performance in corners. With the increased flex, a radial tire also tends to absorb small bumps better, making a smoother ride.
That said, motorcycles are built from the tire up. This means that when radial motorcycle tires came on the market, bike design changed -- new frames, new suspensions, etc. Your bike is designed for bias or radial tires -- not both. Run the tire your bike's made for or check with manufacturers' recommendations before switching.
Above all, don't run bias on one wheel and radial on the other, which can result in an unpredictable ride.