The difference is the “vice-versa”: any can flow into other types but mixed can not.
/* @flow */
var numeric:number = 0;
var anyTyped:any;
var mixTyped:mixed;
numeric = anyTyped;
numeric = mixTyped; // This will throw a flow check error: "Cannot assign `mixTyped` to `numeric` because mixed is incompatible with number. [incompatible-type]"
From the docs you linked to:
It is worth calling out any specifically because of the special nature of this annotation. Use any to escape the static typing of Flow. In other words, if Flow is getting in your way, and you are absolutely convinced your program is type correct, you can silence the errors by annotating locations along the error paths with type any.