Depends on the quality of the SG. In a good one you're part of a social scene, you have direct access to a number of active players which is helpful in setting up a team, organising TFs and PvP excursions etc. You can do SG-only mishes. You get alternate colours for your costume. You get teleporter access for quicker transport. You can join in raids.
Can't think of anything else at the moment...
When your SG is established and has an item of power in your base, your SG gets xp/damage bonuses etc (depending on the actual IoP)
Like people above me have said, more of a social thing. Need to get into a team or help, ask your sg and those its coalitioned with.
Being in SG mode benefits the SG by gaining it prestige and salvage. However, not sure if you are aware but after lv 25 the amount of infamy you receive will decrease. You shouldn't be expected to be in SG mode all the time. Its a juggling act. If the SG says you have to be in sg mode all the time, find a new one.
Heros
Max Powerz: Lv 50 Tanker
Villains
Max Khaos: Lv 50
Overlord of E.V.I.L.
My Brute - http://maxy-khaos.mybrute.com/
LF SG! Arc ID# 193083
And, at 35, INF gain will disapear when in SG mode totaly 
So... at that point, you will NEED to be out of SG mode to earn INF.
I have a question(i tried looking in forum for a sensible answer)what do you actually get back from a SG?....the reason i ask is when in SG mode i lose 161 infamy per kill and 4500 infamy from end of mish, thats a lot of infamy to lose, near on 10k a mission, i'm lvl 32 and would like to equip fully with S/O's but will never afford it at this rate, so all this prestige i'm earning the SG what can i get from it.??? SO enhancements maybe, don't think so.