I think the advice is not meant for scala especially, but for OO in general (so far I know scala is supposed to be a best-of-breed between OO and functional).
Overriding is fine, it's the heart of polymorphism and is central to OO design.
Overloading on the other hand is more problematic. With method overloading it's hard to discern which method will be really invoked and it's indeed a frequently a source of confusion. There is also rarely a justification why overloading is really necessary. The problem can most of the time be solved another way and I agree that overloading is a smell.
Here is an article that explain nicely what I mean with "overloading is a source of confusion", which I think is the prime reason why it's discouraged. It's for java but I think it applies to scala as well.
static overload resolution can rob an API of all type safety:
scala> object O { def apply[T](ts: T*) = (); def apply(f: (String => Int)) = () }
defined object O
scala> O((i: String) => f(i)) // oops, I meant to call the second overload but someone changed the return type of `f` when I wasn't looking...
The reasons that Gilad and Jason (retronym) give are all very good reasons to avoid overloading if possible. Gilad's reasons focus on why overloading is problematic in general, whereas Jason's reasons focus on why it's problematic in the context of other Scala features.
To Jason's list, I would add that overloading interacts poorly with type inference. Consider:
val x = ...
foo(x)
A change in the inferred type of x could alter which foo method gets called. The value of x need not change, just the inferred type of x, which could happen for all sorts of reasons.
For all of the reasons given (and a few more I'm sure I'm forgetting), I think method overloading should be used as sparingly as possible.