What is the effect of Combat Ranks in Elite Dangerous?

I understand that you can increase your Combat Rank (Harmless, Mostly Harmless, etc.) by killing other commanders.

But what does it actually mean? What's the benefit of getting higher ranks?

What is the difference between a commander with an Elite rank compared to a Harmless rank?

Nothing really.

It USED to be that you could only take on certain missions if your combat / trade / exploration rank was xxxx level but now that doesn't affect which missions you can take (it does mean you get a larger payout though)

Really, it's just something nice to show off these days. With NPCs obviously the higher they are the tougher but for commanders it just shows what part of the game they invested their time into.

also, your combat rank goes up regardless of NPC / Commanders. Both will cause it to increase

There's two important effects I'm aware of.

Shinrarta Dezhra

Achieving a rank of Elite in Combat (or Trade, or Exploration, or CQC) will give you access to the Shinrarta Dezhra system.

The Shinrarta Dezhra system is particularly important because it is home to the Jameson Memorial station, which orbits Founders World. Jameson Memorial is most notable for always having all player-purchasable ships and modules available for sale. An added bonus there is that ships and modules are sold at a 10% discount.

It is possible to get access to this system without having an Elite rank, but only if you backed the Kickstarter campaign for the game.

Engineers

Some Engineers require certain Pilots Federation ranks for you to gain access. Those known to have such requirements at this time are:

• Felicity Farseer, requiring Exploration rank Scout or higher.
• Broo Tarquin, requiring Combat rank of Competent or higher.
• Didi Vatermann, requiring Trade rank of Merchant or higher.

Besides allowing you to take on higher tier combat based missions. Your combat rank also helps determine how much exp you earn for every ship you kill. The higher combat rank you have the harder the ships you are going to need to fight to earn exp points.

This wiki page shows a chart on the amount of exp you earn comparing your combat rank to that of the enemy's ship.