Rank is for Major Factions (Empire / Federation / Alliance - there are only three and they control lots of star systems).
Better rank gives access to ship / systems etc. They do not (directly) give out missions. Rank just goes up. You can rank with all major factions, they don't care.
Reputation is Minor Factions (only present in a handful of systems).
They can be allied with a Major Faction. They give out missions through brokers in the Mission board in stations, and they can control stations. Having a better reputation gives access to better missions. Missions can cause you to lose reputation with a minor faction.
I would concentrate of trying to understand the difference between minor and major factions. The System map lists the minor factions and who they are allied with in second tab (I think)
Look up a guide on the Background Simulation for more there is too much to repeat here. I've not found a good one, this one
https://forums.frontier.co.uk/showthread.php/193064-A-Guide-to-Minor-Factions-and-the-Background-Sim talks too much about System States, which isn't what you ask about (but is an important thing that minor factions do). It doesn't help that people interchangeably use rank & reputation - hence my rant - apologies
Green Tentacle from Day of the Tentacle

(I think they were in Maniac Mansion as well)