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I'm sure that these terms should be different since there exists a difference between parameter and argument in computer science but I'm not sure about their differences in math.

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    A *function* has argument-places, like the "sum" function, that has two (binary function). In order to synbolize them, we use variables : $f(x,y)$. – 2017-01-25
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    In principle we may write $f(-,-)$ with place-holders, but then we are in trouble because we cannot make a difference between $f(x,x)$ and $f(x,y)$. – 2017-01-25
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    @MauroALLEGRANZA So, argument is a place and variable is the stuff that fills that place? – 2017-01-25
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    In a mathematical expression, like e.g. $ax+by$ we call $a,b$ *parameters* in order to convey the fact that - in the context of the "discourse" about that expression - we will consider them *constant* while $x,y$ are *variables*. But the "discourse" will holds generally, irrespective of the specific values of $a,b$, and this is why we use letetrs instead of (individual) numbers. – 2017-01-25
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    Not exactly: variables are symbols and they are the place-holders. We assign *values* (input) to variables and "compute" the resulting value (output) of the function (or expression) for those input values. Basically, it is the same as in computer science: we use math formulae in the same way as the computer uses FORTRAN or C code. – 2017-01-25
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    @MauroALLEGRANZA Isn't it is the case that we assign values to arguments too and then do the computation – 2017-01-25
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    Consider the *function* "Father of...": it returns as output value (the name of) a man (the Father) for each input value (the name of) a man. We may write it : $\text {FatherOf}(x)$, where the place-holder has little use. But with the *relation* "--- is Father of..." we need them : $\text {FatherOf}(x,y)$. – 2017-01-25
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    Yes; it is only a question of terminology... The "process" can be described as well with the following pseudo-formula: $+(\text{argument}_1, \text{argument}_2)=\text{output}$. When you have to write a full book, $+(x,y)=z$ is much easier., – 2017-01-25

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Consider family of functions $\{f_a : a\in [0,1]\}$, where $f_a:\mathbb{R}\rightarrow \mathbb{R}$ is given by $$f_a(x)=x^2 - a.$$ When you take some $x\in \mathbb{R}$, this is a variable. This variable becomes an argument for function $f_1$ if you begin to consider $f_1(x)$. In this expression, $1$ is a parameter.

More generally, parameter is a "selector" from a family of "similar" functions. Variable is basically any element from any set. If you take variable from domain of some function to consider value of this function at given point, then variable becomes an argument for the function.

I hope it made this topic more understandable.

Remember, that these terms, though diffeerent, are sometimes used interchangeably. For example, you can consider a function of two variables $f(x,y)$ as a function of one variable $y$ with parameter $x$.