What are the differences in the two possible methods to calculate the Hardy-Weinberg Equation?
Okay, so the Hardy-Weinberg Equilibrium is just that: an equilbrium.


In physics, a system (some object) is at equilibrium when no net energy crosses its boundary with the surroundings.


no energy change means no change in motion (the state of the object) due to outside influence on the system.

In chemistry, a system (the beaker/reaction container/cytosol) is at equilibrium when no change in the concentration of the chemical species inside of it occurs over time


no chemical makeup change means the (molecular) state of the atoms in the system is free of energetic influence from the outside world.

In the H-W Equilibrium, a system (the gene pool) is at rest when no change in the frequency(concentration) of its alleles occurs over time.


no change in the frequency of alleles means that there is no outside influence on the state of the gene pool.



Just like in basic physics, we have to define our system. If our system is the gene pool, then even the behaviors of the organisms in whose cells those alleles are found are outside of the system. The system is only the alleles aka gene pool.

The most important part about understanding the H-W Equilibrium is that it only applies when those conditions of the gene pool being free from outside influence hold. In fact, in the link you gave, the introduction has it in the second sentence (don't feel bad, I missed it the first few times, too—too focused on applying the algebra to understand what it was really saying):


  The model has five basic assumptions:
  
  1) the population is large (i.e., there is no genetic drift)
  
  2) there is no gene flow between populations, from migration or transfer of gametes
  
  3) mutations are negligible
  
  4) individuals are mating randomly
  
  5) natural selection is not operating on the population.


Let's break them down/group them:


2: Obvious in terms of equilibrium == system is in isolation from surroundings
4 &amp; 5: Selection pressure (4: sexual selection, or 5: natural selection) is not pushing on the system.
1 &amp; 3: Random effects are not pushing on the system (1 = statistical sampling error that can push small populations, and 3 = mutation that pushes the system like adding a new chemical species to a system at rest.)



With that solid grounding in the theory, let's move on to your questions about the algebra. Remember, the equilibrium only applies when the gene pool is not being pushed around by external factors. Which means that the physical copies of alleles (gametes) should pair randomly with the possible gametes at every mating. If those assumptions didn't hold true, then the math of the equilibrium wouldn't hold up. If you run the numbers on a system at equilbrium, barring small amounts of sampling error (reducible with a bigger n), the H-W Model's predictions will be accurate. All the methods will yield similar results.

However, if the predicted values don't match the actual with one method, you can assume that other methods will also yield inaccurate predictions, and that the two different calculation methods' predictions will vary widely from each other, largely because of the assumptions you are making (Method 2: 1 = p+q), or aren't making (Method 2).