I understand that each particle is a solution to a specific function, and each particle and the swarm is constantly searching for the best solution. If the global best is found after the first iteration, and no new particles are being added to the mix, shouldn't the loop just quit and the first global best found be the most fitting solution? If this is the case what makes PSO better than just iterating through a list.
Your terminology is a bit off. Simple PSO is a search for a vector x that minimizes some scalar objective function E(x). It does this by creating many candidate vectors. Call them x_i. These are the "particles". They are initialized randomly in both position and rate of change, also called velocity, which is consistent with the idea of a moving particle, even though that particle may have many more than 3 dimensions.
Simple rules describe how the position and velocity change over time. The rules are chosen so that each particle x_i tends randomly to move in directions that reduce E(x_i).
The rules usually involve tracking the "single best x_i value seen so far" and are tuned so that all particles tend to head generally toward that best value with random variations. So the particles swarm like buzzing bees, heading as a group toward a common goal, but with many deviations by individual bees that, over time, cause the common goal to change.
It's unfortunate that some of the literature calls this goal or best particle value seen so far "the global minimum." In optimization, global minimum has a different meaning. A global minimum (there can be more than one when there are "ties" for best) is a value of x that - out of the entire domain of possible x values - produces the unique minimum possible value of E(x).
In no way is PSO guaranteed to find a global minimum. In fact, your question is a bit nonsensical in that one generally never knows when a global minimum has been found. How would you? In most problems you don't even know the gradient of E (which gives the direction taking E to smaller values, i.e. downhill). This is why you are using PSO in the first place. If you know the gradient, you can almost certainly use numerical techniques that will find an answer more quickly than PSO. Without a gradient, you can't even be sure you've found a local minimum, let alone a global one.
Rather, the best you can usually do is "guess" when a local minimum has been found. You do this by letting the system run while watching how often and by how much the "best particle seen so far" is being updated. When the changes become infrequent and/or small, you declare victory.
Another way of putting this is that PSO is used on problems where reducing E(x) is always good and "you'll take anything you can get" regardless of whether you have any confidence that what you got is the best possible. E.g. you're Walmart and any way of locating your stores that saves/makes more dollars is interesting.
With all this as background, let's recap your specific questions:
If the global best is found after the first iteration, and no new particles are being added to the mix, shouldn't the loop just quit and the first global best found be the most fitting solution?
There's no answer because there's no way to determine a global best has been found. The swarm of buzzing particles might find a new best in the next iteration or ten trillion iterations from now. You seldom know.
If this is the case what makes PSO better than just iterating through a list?
I don't exactly grok what you mean by this. The PSO is emulating the way swarms of biological entities like bugs and herd animals behave. In this manner it resembles genetic algorithms, simulated annealing, neural networks, and other families of solution finders that use the following logic: Nature, both physical and biological, has known-good optimization processes. Let's take advantage of them and do our best to emulate them in software. We are using nature to do better than any simple iteration we might devise ourselves.