Ernest, I am not a photography guy. I have done enough research to believe I have a handle on how it works and how it relates to thermal tech.
50mm Objective size with f/1.0 lens = 50mm focal length
50mm Objective size with a f/1.2 lens = 60mm focal length
42mm Objective size with a f/1.2 lens = 50mm focal length
The first and third example have the same ending focal length. However the 3rd example needed a smaller objective size (less Germanium needed for a thermal scope) to get to the same amount of magnification.
Most companies list the focal length because this is what determines the FOV and the magnification in accordance with the micron and resolution parameters.
The higher (longer) the focal length value, the more base magnification and the smaller the FOV.
So, the basic answer to your question is the difference between a 1.0 lens and a 1.2 lens has a higher (longer) focal length (if their objective diameter is equal which translates in to less FOV and higher base magnification with all other parameters being equal.
Pulsar kind of writes their spec in an odd way, but I believe in writing it as F50/1.0 and a F50/1.2,they will both have a 50mm focal length. However, the 1.0 will have a 50mm objective and the 1.2 will have a 42mm objective.
To add to the confusion the new Thermion 2 XQ38 has a 35mm focal length and a f/1.0 rating. So, the 38 has nothing to do with 38mm as according to their specs it has a 35mm objective size and a 35mm focal length, but they call it an XQ38.