In this paper, we propose a new approach to formulating the subresultant polynomials for two Bernstein polynomials and develop two explicit formulas for them, i.e., one in the determinant form and the other in the determinental polynomial form. It should be pointed out that both formulas can be expanded into polynomials in the Bernstein form, and the resulting subresultant polynomials are exactly the same as those for the polynomials obtained by expanding the given Bernstein polynomials into their standard power-basis forms. In addition, two applications are provided to show the effectiveness of the newly developed subresultant formulas, i.e., computing the greatest common divisor of parametric Bernstein polynomials and solving the real root classification problem for Bernstein polynomials in the unit interval.