I have been doing some research into computational chemistry, having never looked into the area before, and have read about a popular, paid-for software "Gaussian", and free alternatives such as "GAMESS". From what I understand they are both capable of employing methodologies such as PM3, but I am struggling to understand the point of the software itself; how is a Gaussian PM3 calculation any different to a GAMESS calculation? There must be a reason one would want to pay for it.
There are several interrelated reasons (disclaimer: I am a GAMESS developer).
Gaussian has a lot of features and interoperability. To take your PM3 example. Yes, the results of a PM3 calculation are identical in GAMESS and Gaussian. But until recently you couldn't do a PM3 calculation in solution using the PCM method in GAMESS. Also, what if you want to do a PM6 calculation instead. In GAMESS this is currently only possible only for elements up to F.
Because of this Gaussian has been the program of choice for many people, which also means they have trained a lot of people to use Gaussian. Learning to use a new (and perhaps free) program well can be very time consuming, so it might be more cost effective to pay for a program you know how to use well. And chances are you probably have to learn how to use several new programs to get all the functionality of Gaussian.