Is Probability Mass Function essentially the same as a list of Probabilities? For the most common example of tossing a coin two times. There are four outcomes.
We know that range of X = Rx{0,1,2} Where X is the event of an outcome of heads at least once.
Px(0) = 1/4 two tails and no heads
Px(1) = 1/2 HT and TH
Px(2) = 1/4 Two heads and no tails
The limitations is that Px(k) > 0 and ΣPx = 1.
Is there something more to PMF than being a list of probabilities? If no then why is there an entire equation making it such?
 A: Yes, a p.m.f. is just a list of probabilities, if you allow lists of infinite length.  But not all random variables have a p.m.f., so we care about when they exist!
For an example where infinite length is required, take $X\sim\mathrm{Geo}\left(\frac{1}{2}\right)$.  Then for any $x\in\mathbb{N}$, $$\mathbb{P}[{X=x}]=\frac{1}{2^x}$$
For an example with no p.m.f., suppose $X\sim\mathcal{U}([0,1])$; that is, $$\mathbb{P}[{a\leq X\leq b}]=\min{\!(b,1)}-\max{\!(a,0)}\quad\quad\quad(a\leq b)$$  For any $x\in[0,1]$, $$\mathbb{P}[{x=x}]=\mathbb{P}[{x\leq X\leq x}]=\min{\!(x,1)}-\max{\!(x,0)}=x-x=0$$  So a "list" of probabilities describing the possible values of $X$ would just be  a list of zeroes.  That isn't very useful, because it's impossible to deduce from the "p.m.f." such facts as $$\mathbb{P}[{0\leq X\leq 1}]=2\mathbb{P}\left[{0\leq X\leq\frac{1}{2}}\right]$$
A: That question is pseudo-ambiguous so its difficult to answer as being yes or no. To answer that question, one needs to be clear about what a probability space is and what a random variable is. This is because the pmf will depend on the random variable. You can have several different random variables defined on the same sample space and each one will generally have a different pmf. A sample space is difficult to define but will generally be either the population that you are interested in (or sample thereof) or the set of potential outcomes of some experiment. So if your experiment is a dice roll than your sample space will be $\{1,2,3,4,5,6\},$ which corresponds to the possible outcomes of that dice roll.
An example of a population-oriented one would be if you are studying how the citizens of a country between the ages of 50 and 65 feel about each candidate running for political office, than your sample space would be the set of all citizens of said country that are between the ages of 50 and 65, or perhaps a subset of that population that you have called and interviewed (i.e. a sample). Its easier to get a feel for these concepts by taking the experimental perspective of a sample space, so I will focus on that.
A subset of the sample space is called an "event", and an event is said to have "occured" if any of its members occurs. So for example, with the dice roll the event corresponding to "the die will land on an even number" would be $\{2,4,6\},$ and so that event will have occured if the die lands on any of those three numbers. A "probability measure" is a function that assigns probabilities to different events, in such a way that the entire sample space is assigned a probability of 1 (i.e. a 100% chance that one of the outcomes will occur) as well as if two events have no members in common, the probability of the union of those events, which will itself be an event, will equal the sum of the probabilites of those events.
Generally you think of the union of two events as corresponding to the word "or". So a probability measure assigns probabilities to events in such a way that if two events A and B are mutually-exclusive (i.e. cannot both occur), than the probability of the event "A or B" will be equal to the probability of A plus the probability of B. These three things, a sample space, a collection of events, and a probability measure together form what is called a probability space. So strictly speaking, it is actually the probability measure that assigns the probabilities to each outcome.
A random variable is a function that assigns a real number to each element of a sample space, and it is possible (and will generally be the case) that you will have multiple sample points assigned the same number. Using the above example, one random variable might be to assign the number 0 to 1,3,5 and assign the number 1 to 2,4,6. (basically assigning 0 the sides of the die that are odd and 1 to the sides that are even). The purpose of a random variable is to allow you to represent the sample space members as numbers so that you can manipulate them numerically.
A pmf is a function whose domain is the range of that random variable and its value is equal to the probability of the event consisting of all members of the sample space that are mapped by the random variable to that number. Symbolically, if $P(x)$ is my probability measure, and $X(x)$ is my random variable, than the pmf, $f(x)$, will be defined as $f(x):= (P \, \circ X^{-1})(x) $, thought of as the probability that $X$ will equal $x$ (i.e. that the outcome will be one of the sample points that is mapped to $x$ by $X$). For the above example, the range of my random variable is $\{0,1 \}$. I would likely have assigned the probabilities corresponding to those events as 0.5 to each (that is 50% chance that an even number will be rolled and 50% chance that an odd number will be rolled), so my pmf would be f(0) = 0.5 and f(1) = 0.5.
The "limitations" you have spoken to basically state that a probability must always be positive and that the probabilities of the possible outcomes must sum to 100% (equivalently, that there is a 100% probability that at least one of the potential outcomes/events will occur, and it is a consequence of the fact that the probability measure $P$ must satisfy those constraints. Generally your random variable will partition the sample space into the events that you are interested in. If you are interested in 5 different potential events than your random variable will assign every member of the sample space to one of 5 different numbers. In the above instance we were interested in whether the outcome would be even or odd, and I assigned all even numbers the same number (1) as well as all odd numbers the same number (0). From this perspective, your pmf will effectively be a function whose input is a numerical representative of an event that interests you and its value will be the probability that event occurs. Instead of looking at whether the outcome of the dice roll was even or odd, I could have been interested in whether the roll would have landed on a prime number, in which case I would have had a different random variable and consequently a differen t pmf.
So the answer is both yes and no. A pmf provides a list of probabilities, but its value is tied to the random variable you are looking at, whereas the probability measure for the sample space assigns probabilities in an absolute way.
