What are the differences between functions and methods in Swift?

As you said yourself, methods are functions, but in a class. In objective-c you never realized this, because we were only coding in classes. Every function that we wrote was a method of a class (ViewController or some other class we created).

In Swift we have the ability to create functions that are not inside some class. The main reason for doing this is to write functions that are not tied to any class, and can be used wherever we need them. So if you have a function that is related to a class you write it inside the class and you can access is from every instance of the class:

class Square {
   var length: Double
   func area() -> Double {
      return length * length
   }
}

But if you need to access the function from everywhere, then you don't write it inside a class. For example:

func squared(number: Int) -> Int {
    return number * number
}

About your syntax issues between functions and methods: You guessed it right, methods and functions are called a little bit differently. That is because in Objective-C we had long method names and we liked them because we could read what the methods were doing and what the parameters were for. So the first parameter in a method is in most cases described by the function name itself. And the other parameters shouldn't only be some numbers or strings or instances, they should be described as well, so Swift writes the name of the variable automatically. If you want to describe it by yourself you can do that as well:

class Something {
    func desc(firstString string1: String, secondString string2:String) {...}
}

After a few hours of reading and experimenting, here are the things that I found out:-

Functions in Swift

Functions are self-contained chunks of code that perform a specific task. You give a function a name that identifies what it does, and this name is used to “call” the function to perform its task when needed.

Resource: Official Apple Documentation on Functions in Swift

Function Parameter Names

However, these parameter names are only used within the body of the function itself, and cannot be used when calling the function. These kinds of parameter names are known as local parameter names, because they are only available for use within the function’s body.

It means that by default, all the parameters for Function are local parameters.

But, sometimes we want to indicate the purpose of each parameter. So, we can actually define an external parameter name for each parameter. Example Code:

func someFunction(externalParameterName localParameterName: Int) {
    // function body goes here, and can use localParameterName
    // to refer to the argument value for that parameter
}

Another way to make the external parameter name is using hash symbol (#) to shorten the name.

func someFunction(#localParameterName: Int) {
    // function body goes here, and can use localParameterName
    // to refer to the argument value for that parameter
}

To call the above functions with external parameter, you may use

someFunction(localParameterName:10)

Methods in Swift

Methods are functions that are associated with a particular type. Classes, structures, and enumerations can all define instance methods, which encapsulate specific tasks and functionality for working with an instance of a given type.

Resource: Official Apple Documentation on Methods in Swift

However, the default behavior of local names and external names is different for functions and methods.

Specifically, Swift gives the first parameter name in a method a local parameter name by default, and gives the second and subsequent parameter names both local and external parameter names by default.

Code below shows the differences for default and non-default parameters for method in Swift.

import Foundation
import UIKit

class ViewController2: UIViewController {
    override func viewDidLoad() {
        super.viewDidLoad()

        //Default methods calling
        var dailyStatement = greet("Rick", day: "Tuesday")
        println(dailyStatement)

        //First parameter is also an external parameter
        var dailyStatement2 = greet2(name:"John", day: "Sunday")
        println(dailyStatement2)
    }

    //Default: First Parameter is the local parameter, the rest are external parameters
    func greet (name: String, day: String) -> String {
        return "Hello \(name), today is \(day)."
    }

    //Use Hash symbol to make the First parameter as external parameter
    func greet2 (#name: String, day: String) -> String {
        return "Hello \(name), today is \(day)."
    }
}

I might miss some important details. Hope someone can provide a better answer.

Well, @Ricky's answer says it pretty much. I was confused what exactly they are. So here is my thought:

Functions could be defined outside of classes or inside of classes/structs/enums, while Methods have to be defined inside of and part of classes/structs/enums.

We could define a Function outside of any Type's definition and could use it within Methods of any Type's definition.

Just my understanding and illustration here, hope this helps someone else or you may edit if you feel there is an improvement needed OR let me know if anything is wrong:

//This is a Function which prints a greeting message based on the category defined in an 'enum'
func greet(yourName name: String, category: GreetingsCategory) {
    switch  category {
        case .Person:
            print("Hello, " + name + " Today is Tuesday")
        case .Vehicle:
            print("Hello, " + name + " your Vehicle is a Car")
    }
}

//This is an 'enum' for greetings categories
enum GreetingsCategory: String {
    case Person
    case Vehicle
}

//Type: Person
class Person {

    //This is a method which acts only on Person type
    func personGreeting() {
        greet(yourName: "Santosh", category: .Person)
    }
}

//Type: Vehicle
class Vehicle {

    //This is a method which acts only on Vehicle type
    func vehicleGreeting() {
        greet(yourName: "Santosh", category: .Vehicle)
    }
}

//Now making use of our Function defined above by calling methods of defferent types.
let aPerson = Person()
aPerson.personGreeting()
//prints : Hello, Santosh Today is Tuesday

let aVehicle = Vehicle()
aVehicle.vehicleGreeting()
//prints: Hello, Santosh your Vehicle is a Car

//We can also call the above function directly
greet(yourName: "Santosh", category: .Person)