You call them "local" factions, but I prefer "minor" factions, since they can have a presence in multiple systems.
Benefits include:
- You get friendly/allied with the major factions by becoming friendly/allied with multiple of the minor factions allied with them. Being friendly/allied with major factions makes you automatically friendly with all of their minor factions, and has other benefits.
- Lower fuel and repair costs at any station they control.
- More/Better missions on the bulletin board. If you look at the bulletin board, you can see that missions require a minimum reputation with the faction posting the mission, and some require "friendly" or "allied".
- In a few cases, it's how you gain a permit to a system. There's a bunch of system permits tied to military rank with the Federation and Empire, but there's also systems controlled by minor factions that you have to get friendly or allied with to get the permit.
- Less attention from their system authority (police) ships. Less likely to be interdicted, slower to get scanned on your way into a station, etc. This makes travel in that system easier, and especially makes smuggling easier.
- It's further from "Unfriendly" and "Hostile". Those have definite disadvantages, and you can get away with more things before you get there if you start at friendly or allied.
- Their ships show up green on your scanner. Depending on what you're doing, this can be helpful. I find bounty hunting in a friendly/allied system easier, since I can skip scanning green ships.
- You get nicer messages from the station when you request docking.
If you rank up from "Friendly" to "Allied", it's basically just more of the same types of things.
As of Elite: Dangerous 1.3, players can pledge themselves to a galactic power of their choice. The galactic powers are smaller in scale than the galactic superpowers, and are often parts of the government of a superpower, but not always. For example, Felicia Winters is the Shadow President of the Federation.
In addition to advancing the political agendas of your favorite political figure of the Milky Way, players recieve some rewards for participating in powerplay. For example, players pledged to Emperor Arissa Lavigny Duval recieve a bonus to bounty payout, and can purchase a burst railgun module at rank 3.
There is a cost to participating in powerplay, however: you are marked as being pledged to that power. So if you're pledged to the Emperor, and find yourself in Federation space, you will be marked as an enemy, and anyone will be allowed to interdict and attack you without receiving a bounty. This works the other way around, though: if you are in exploited space, and see an enemy ship, you are allowed to interdict and destroy them without consequence.
For more information about all of the powers, check out the Elite: Dangerous Wiki.
Best Answer
While there's some consideration towards (re)introducing interstellar bounties, you don't have to worry about your reputation preceding you across the galaxy. Because bounties are now scoped to the system jurisdiction, the only time you'll ever get in trouble with the law for a given bounty is within that particular system, regardless of their major faction association.
When you perform an illegal action that triggers a bounty, there'll be a cooldown period of up to seven days in which the bounty will be visible to that system's authority, as well as commanders scanning you with a Kill Warrant scanner. Committing additional bounty-level offenses in the same system will both increase the bounty amount and reset the timer.
Once the time expires, the bounty will become dormant for another seven days, during which time it'll only be visible if the system authority for the issuing system scans you. If this happens, the bounty will become active again.
Bounties that also complete the dormant cooldown become legacy fines, which are a fee that's added to your rebuy cost if you happen to have to respawn in a station or outpost owned by the issuing faction. These never expire, but can be paid off via the Contacts menu in a station/outpost within the issuing jurisdiction*.
*I think paying them off might have some minorly positive effect on your reputation with the faction as well, but I haven't Scienced™ it.