Question: I've heard and read about how recording/mixing engineers in the music business analyze each others mixes to learn the techniques used to make a certain track sound so good or what the latest trend/style is being used for them to incorporate in their work. They listen to how the track is balanced, panned and where each instrument fits in the frequency spectrum and so on, I'm guessing the same applies to a film or video game?, what else is there to look for?

Answer: 
The music engineering world is very different from a technical standpoint because the music usually stands on its own, giving the engineers, producers and artists free reign (more or less) to make choices without penalty. On the other hand, film and game sound design must work harmoniously with visual elements to tell a story (as Hubert described above). It's quite easy to get in the habit of listening to the finer details of the mix (panning, processing, reverbs, etc.), but remember that the ordinary listener appreciates sound design as a whole gesture and not as a sum of parts. Moreover, many designers strive to design that way too, trumping slight imperfections in the mix with contextual intent.

When watching films and playing games, sit back, turn off the lobe of your brain that says "whoa what's the reverb tail on that", and just enjoy the experience first. Try to listen without expectations, and then when you listen again and check out the detail you'll have a better understanding of why the designers made the decisions they did.
I found it effective to use Adobe Audition and XMind (mind mapping tool). 

In Audition I am able to see both waveform and spectrum at the same time. I set markers at the in and out points of certain audio sections that I'm interested in and then use Marque Tool to listen to exact frequency range of the sound. I can reverse, de-noise, make it louder. 

In XMind I have a template mindmap for this kind of task with several sections that I need to fill in, like Melody, Harmony, Rhythm, Instruments, FX, Structure, Mix, Automation, Stats etc. I tried to use simple lists and spreadsheets for this but I'm a visual person and mindmaps just work better for me.