Why do we need a Probability Mass Function? We know that $P(X=x) = p_X(x)$
For instance, let $X$ = number of heads if two fair coins are tossed simultaneously, and $TT = 0, HT=TH=1, HH=2$.
Then we have: $P(X=1) = ( \frac{1}{4} + \frac{1}{4} ) = \frac{1}{2}$. 
The same information is being conveyed by a PMF as $p_X(1) = \frac{1}{2}$.
So, why do need a PMF? Why isn't a Probability function enough?
 A: For a random variable $X$ that takes a finite number of values, the "probability mass function" (probability that $X$ takes each of the possible values) is indeed enough for computing the probability of any event. (And often the expressions "probability mass function" and "probability function" are used interchangeably in this context).
This is also true for a discrete random variable (i.e. a variable that takes values over a finite or countably finite support, e.g, the integers).
But it's not true for other variables. Namely, for a continuous variable that takes values over all reals (say, a uniform over the interval $[0,1]$). Here the pmf doesn't make sense, because $P(X=x)=0$ for all $x$. 
An alternative is the cumulative distribution function (CDF) that is well defined for a real random variable, no matter it's discrete or continuous (or anything), and fully characterizes the probability distribution. If the CDF is derivable, the derivative is called the probability density function - this is conceptually analogous (but not the same) as the PMF , for continuous variables.
A: In the comments, you clarify that a probability function is the function taking an event to its probability. You are correct that a probability function is enough to recover the PMF. However, for a discrete random variable, the probability function has lots of redundant data (given by the axioms of probability), whereas the PMF does not.
For example, let $X$ be the outcome when a weighted die is rolled. The events are the subsets of $\{1,2,3,4,5,6\}$. There are $64$ events, so it might seem like we would have to give $64$ values to describe the probability function. However, the PMF only takes on $6$ values (the probabilities of rolling each individual number), and describing these $6$ values is enough to recover the probability function. So it is much faster to write down the PMF than to write down the probability function.
