I would argue that typeclasses are an essential part of Haskell.
They are the part of Haskell that makes it the easiest language I know of to refactor, and they are a great asset to your being able to reason about the correctness of code.
So, let's talk about dictionary passing.
Now, any sort of dictionary passing is a big improvement in the state of affairs in traditional object oriented languages. We know how to do OOP with vtables in C++. However, the vtable is 'part of the object' in OOP languages. Fusing the vtable with the object forces your code into a form where you have a rigid discipline about who can extend the core types with new features, its really only the original author of the class who has to incorporate all the things others want to bake into their type. This leads to "lava flow code" and all sorts of other design antipatterns, etc.
Languages like C# give you the ability to hack in extension methods to fake new stuff, and "traits" in languages like scala and multiple inheritance in other languages let you delegate some of the work as well, but they are partial solutions.
When you split the vtable from the objects they manipulate you get a heady rush of power. You can now pass them around wherever you want, but then of course you need to name them and talk about them. The ML discipline around modules / functors and the explicit dictionary passing style take this approach.
Typeclasses take a slightly different tack. We rely on uniqueness of a typeclass instance for a given type and it is in large part it is this choice permits us to get away with such simple core data types.
Why?
Because we can move the use of the dictionaries to the use sites, and don't have to carry them around with the data types and we can rely upon the fact that when we do so nothing has changed about the behavior of the code.
Mechanical translation of the code to more complex manually passed dictionaries loses the uniqueness of such a dictionary at a given type. Passing the dictionaries in at different points in your program now leads to programs with greatly differing behavior. You may or may not have to remember the dictionaries your data type was constructed with, and woe betide you if you want to have conditional behavior based on what your arguments are.
For simple examples like Set you can get away with a manual dictionary translation. The price doesn't seem so high. You have to bake in the dictionary for, say, how you want to sort the Set when you make the object and then insert/lookup, would just preserve your choice. This might be a cost you can bear. When you union two Sets now, of course, its up in the air which ordering you get. Maybe you take the smaller and insert it into the larger, but then the ordering would change willy nilly, so instead you have to take say, the left and always insert it into the right, or document this haphazard behavior. You're now being forced into suboptimal performing solutions in the interest of 'flexibility'.
But Set is a trivial example. There you might bake an index into the type about which instance it was you are using, there is only one class involved. What happens when you want more complex behavior? One of the things we do with Haskell is work with monad transformers. Now you have lots of instances floating around -- and you don't have a good place to store them all, MonadReader, MonadWriter, MonadState, etc. may all apply.. conditionally, based on the underlying monad. what happens when you hoist and swap it out and now different things may or may not apply?
Carrying around an explicit dictionaries for this is a lot of work, there isn't a good place to store them and you are asking users to adopt a global program transformation to adopt this practice.
These are the things that typeclasses make effortless.
Do I believe you should use them for everything?
Not by a long shot.
But I can't agree with the other replies here that they are inessential to Haskell.
Haskell is the only language that supplies them and they are critical to at least my ability to think in this language, and are a huge part of why I consider Haskell home.
I do agree with a few things here, use typeclasses when there are laws and when the choice is unambiguous.
I'd challenge however, that if you don't have laws or if the choice isn't unambiguous, you may not know enough about how to model the problem domain, and should be seeking something for which you can fit it into the typeclass mold, possibly even into existing abstractions -- and when you finally find that solution, you'll find you can easily reuse it.
