Any answer which limits this to a specific language is inherently flawed. In addition you must also deal effectively with static methods and subroutines.
Computer science began with the term 'subroutine'. Small sections of repeatable code which could be executed arbitrarily to perform a common action. Examples are found in early programming languages such as BASIC.
Functions were the evolution of subroutines. They take arguments and may or may not return a value. They take some concepts from maths - input, translated to a given output.
With objects we need to be able to call actions on objects and we do this be exposing methods. Like functions they take arguments and may or may not return a value.
Static methods are designed to act on all possible objects of a class.
The problem is that, pure object-orientated programming leaves no scope for the definition of functions (or indeed subroutines). And languages that evolve to become object orientated often retain syntax from functions to implement methods.
In Java we resort to using 'Utility' classes to provide functions as public static
methods. The Math
class in JavaScript is another example of this.
In PHP we tolerate the use of the word function
to define methods.
In C++ we see both functions and methods, neither demarcated. Indeed, C++ makes no reference to methods, calling them member functions.