Nothing. Federal and Imperial civil naval ranks are purely ornamental/honorary. You can be both king and admiral at the same time. Your reputation with local minor factions is completely unaffected (although gaining rep with one faction might make you lose rep with another faction if the mission was against that other faction).
It can be slightly different if you go for PowerPlay, i.e. pledge to one of the powers (not Fed/Imp, but people like Hudson or Duval). In that case, agents of enemy powers might try to attack you(r ship) if you show up on their turf.
Point in case: I am currently not engaging in powerplay. I am, though, Rear Admiral in the Fed auxiliary navy and (right now) Duke in the Imperial auxiliary navy. Right now, I'm flying a Federal Corvette - and doing evacuation missions form the bombed Imperial station in Paresa, collecting money and Imperial rank. The only ship thaqt attacked me was some deluded player - right on top of the rescue ship. They didn't last long after that rescue ship opened fire with its local defense lasers

(to be fair, the first thing I noticed from the attack was my covas' warning about "thermal signature increased").