It’s difficult to get the embedded videos to be responsive and not have any extra black on them anywhere because we lose the ability to input exact dimensions (notice the site you’re linking to isn’t responsive). So this is going to depend on the actual dimensions of your video (more specifically the aspect ratio resulting from those dimensions). It is possible that the ratio of the YouTube video vs that of the Vimeo video you’re using is not exactly the same.
But outside of the actual embedded video itself, the aesthetic design surrounding the slider the site you’re linking to is not going to look the same as the Akita theme’s. Can I see a link to what you’ve setup?