The little achievement pop-up isn't the only thing that's changed on the Xbox One. Actually viewing Xbox One achievements themselves will be better, too.
The first image in this post, which comes from Microsoft's Mike Ybarra, may look like a wallpaper, but it's actually what a specific achievement in Ryse looks like. It's the screen you'd see if you pressed and held the Xbox button when the achievement icon pops up (or the screen you'd see if you looked at this specific achievement later). Basically, the Xbox One gives you a nice visual you can look back on with each achievement, which is pretty cool.
And this is what the screen containing all your achievements for specific games will look like — the left side will show you what percentage of achievements you've gotten in a game, and you can scroll through the pictures to see individual achievements in that game.
How do you like those achievements! pic.twitter.com/X75KpiyEFN
— Mike Ybarra (@XboxQwik) November 16, 2013
Choosing any of those pictures will take you to a screen similar to the first one in this post.
Between this and having the ability to choose the colour of your achievement pop-up, achievements on the Xbox One are a definite improvement over those on the 360.