What is the difference between traits in Rust and typeclasses in Haskell?

At the basic level, there's not much difference, but they're still there.

Haskell describes functions or values defined in a typeclass as 'methods', just as traits describe OOP methods in the objects they enclose. However, Haskell deals with these differently, treating them as individual values rather than pinning them to an object as OOP would lead one to do. This is about the most obvious surface-level difference there is.

The one thing that Rust could not do for a while was higher-order typed traits, such as the infamous Functor and Monad typeclasses.

This means that Rust traits could only describe what's often called a 'concrete type', in other words, one without a generic argument. Haskell from the start could make higher-order typeclasses which use types similar to how higher-order functions use other functions: using one to describe another. For a period of time this was not possible in Rust, but since associated items have been implemented, such traits have become commonplace and idiomatic.

So if we ignore extensions, they are not exactly the same, but each can approximate what the other can do.

It is also mentionable, as said in the comments, that GHC (Haskell's principal compiler) supports further options for typeclasses, including multi-parameter (i.e. many types involved) typeclasses, and functional dependencies, a lovely option that allows for type-level computations, and leads on to type families. To my knowledge, Rust has neither funDeps or type families, though it may in the future.†

All in all, traits and typeclasses have fundamental differences, which due to the way they interact, make them act and seem quite similar in the end.

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† A nice article on Haskell's typeclasses (including higher-typed ones) can be found here, and the Rust by Example chapter on traits may be found here

I think the current answers overlook the most fundamental differences between Rust traits and Haskell type classes. These differences have to do with the way traits are related to object oriented language constructs. For information about this, see the Rust book.

1. A trait declaration creates a trait type. This means that you can declare variables of such a type (or rather, references of the type). You can also use trait types as parameters on function, struct fields and type parameter instantiations.

   A trait reference variable can at runtime contain objects of different types, as long as the runtime type of the referenced object implements the trait.

   // The shape variable might contain a Square or a Circle, 
   // we don't know until runtime
   let shape: &Shape = get_unknown_shape();
   
   // Might contain different kinds of shapes at the same time
   let shapes: Vec<&Shape> = get_shapes();
   

   This is not how type classes work. Type classes create no types, so you can't declare variables with the class name. Type classes act as bounds on type parameters, but the type parameters must be instantiated with a concrete type, not the type class itself.

   You can not have a list of different things of different types which implement the same type class. (Instead, existential types are used in Haskell to express a similar thing.) ^{Note 1}

2. Trait methods can be dynamically dispatched. This is strongly related to the things that are described in the section above.

   Dynamic dispatch means that the runtime type of the object a reference points is used to determine which method that is called though the reference.

   let shape: &Shape = get_unknown_shape();
   
   // This calls a method, which might be Square.area or
   // Circle.area depending on the runtime type of shape
   print!("Area: {}", shape.area());
   

   Again, existential types are used for this in Haskell.

In Conclusion

It seems to me like traits are in many aspects the same concept as type classes. It addition, they have the functionality of object oriented interfaces.

On the other hand Haskell's type classes are more advanced. Haskell has for example higher-kinded types and extension like multi-parameter type classes.

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Note 1: Recent versions of Rust have an update to differentiate the usage of trait names as types and the usage of trait names as bounds. In a trait type the name is prefixed by the dyn keyword. See for example this answer for more information.