I'd suggest that interfaces are different from mixins, in that they don't provide any code. (Yeah, they could, but when I use the word "interface", people familiar with Java flip a switch in their heads and they think that Java's terrible, terrible, almost completely useless implementation of the idea is what I want. It's not.)
The whole point of roles in Perl 6 is two-fold.
First, as you point out, there's a lot of behavior shared vertically between classes that doesn't make sense to fit into a hierarchy. It'd sure be nice to share that code without having to copy and paste it or do complicated things to your inheritance tree.
Second, it's nice to have meaningful names attached to that behavior that you can use to identify the capabilities of objects. This makes generic programming much, much easier (which I contend is the real point of OO). If you want to print something, you care that it can stringify, not that it is or inherits from String somewhere.