The changes in context make the women's voices in Looking for Richard have much more agency than the characters in Richard III.
Richard III doesn't have a lot of female characters who aren't negatively affected by King Richard. Instead, they're tools for him to use and people who are hurt by his actions. Richard even kills his own wife to get a new one; he married the prince's widow, Anne, to help him get his position. He doesn't have much regard for women and mostly uses them.
Looking for Richard is a different story. It's a story about Shakespeare adaptations on stage and in films. The director interviews a lot of actors and actresses about their feelings and thoughts on Shakespeare. The female actresses don't seem any less important than the male actors. Instead, they seem to have an equal voice and agency.
The main difference is that the movie was made by a modern team while the play was written by an author who was a product of his time. Women didn't have as much agency as men—even though Shakespeare wrote many active, important women. Richard III just didn't have that many, and the ones present in the play are relatively unimportant outside of their relationship with Richard. The ones in the movie have opinions and are as important as anyone else. Since the women are just as important, their voices in the movie are more valued than the voices of women in the play.