What's the difference between multiple parameters lists and multiple parameters per list in Scala?

In Scala one can write (curried?) functions like this

def curriedFunc(arg1: Int) (arg2: String) = { ... }

What is the difference between the above curriedFunc function definition with two parameters lists and functions with multiple parameters in a single parameter list:

def curriedFunc(arg1: Int, arg2: String) = { ... }

From a mathematical point of view this is (curriedFunc(x))(y) and curriedFunc(x,y) but I can write def sum(x) (y) = x + y and the same will be def sum2(x, y) = x + y

I know only one difference - this is partially applied functions. But both ways are equivalent for me.

Are there any other differences?

Strictly speaking, this is not a curried function, but a method with multiple argument lists, although admittedly it looks like a function.

As you said, the multiple arguments lists allow the method to be used in the place of a partially applied function. (Sorry for the generally silly examples I use)

object NonCurr {
  def tabulate[A](n: Int, fun: Int => A) = IndexedSeq.tabulate(n)(fun)
}

NonCurr.tabulate[Double](10, _)            // not possible
val x = IndexedSeq.tabulate[Double](10) _  // possible. x is Function1 now
x(math.exp(_))                             // complete the application

Another benefit is that you can use curly braces instead of parenthesis which looks nice if the second argument list consists of a single function, or thunk. E.g.

NonCurr.tabulate(10, { i => val j = util.Random.nextInt(i + 1); i - i % 2 })

versus

IndexedSeq.tabulate(10) { i =>
  val j = util.Random.nextInt(i + 1)
  i - i % 2
}

Or for the thunk:

IndexedSeq.fill(10) {
  println("debug: operating the random number generator")
  util.Random.nextInt(99)
}

Another advantage is, you can refer to arguments of a previous argument list for defining default argument values (although you could also say it's a disadvantage that you cannot do that in single list :)

// again I'm not very creative with the example, so forgive me
def doSomething(f: java.io.File)(modDate: Long = f.lastModified) = ???

Finally, there are three other application in an answer to related post https://stackoverflow.com/questions/4684185/why-does-scala-provide-both-multiple-parameters-lists-and-multiple-parameters-per . I will just copy them here, but the credit goes to Knut Arne Vedaa, Kevin Wright, and extempore.

First: you can have multiple var args:

def foo(as: Int*)(bs: Int*)(cs: Int*) = as.sum * bs.sum * cs.sum

...which would not be possible in a single argument list.

Second, it aids the type inference:

def foo[T](a: T, b: T)(op: (T,T) => T) = op(a, b)
foo(1, 2){_ + _}   // compiler can infer the type of the op function

def foo2[T](a: T, b: T, op: (T,T) => T) = op(a, b)
foo2(1, 2, _ + _)  // compiler too stupid, unfortunately

And last, this is the only way you can have implicit and non implicit args, as implicit is a modifier for a whole argument list:

def gaga [A](x: A)(implicit mf: Manifest[A]) = ???   // ok
def gaga2[A](x: A, implicit mf: Manifest[A]) = ???   // not possible

There's another difference that was not covered by 0__'s excellent answer: default parameters. A parameter from one parameter list can be used when computing the default in another parameter list, but not in the same one.

For example:

def f(x: Int, y: Int = x * 2) = x + y // not valid
def g(x: Int)(y: Int = x * 2) = x + y // valid

That's the whole point, is that the curried and uncurried forms are equivalent! As others have pointed out, one or the other form can be syntactically more convenient to work with depending on the situation, and that is the only reason to prefer one over the other.

It's important to understand that even if Scala didn't have special syntax for declaring curried functions, you could still construct them; this is just a mathematical inevitability once you have the ability to create functions which return functions.

To demonstrate this, imagine that the def foo(a)(b)(c) = {...} syntax didn't exist. Then you could still achieve the exact same thing like so: def foo(a) = (b) => (c) => {...}.

Like many features in Scala, this is just a syntactic convenience for doing something that would be possible anyway, but with slightly more verbosity.

The two forms are isomorphic. The main difference is that curried functions are easier to apply partially, while non-curried functions have slightly nicer syntax, at least in Scala.