Industry - faster unit building, more trade output, happier populace.
Diplo-trade - better trade efficiency, longer trade agreements, more consecutive agreements, more merchants.
Trading:
Send merchants to compete for shares in a center of trade.
Each month your merchants will compete for a share based on trade efficiency.
You gain money per share based on the center of trades product amount.
Diplomacy:
Spheres of influence are obtainable based on your max sphere limit
Your relations to a nation is what determines if they will accept
Spheres give you money (% based on diplo-trade) and will be your allies in war.
Spheres can not have spheres of their own, a person with a sphere can not be made a sphere.
Battle for a sphere of influence happens through diplomats, where you try to break down the influence of the major players and gain the most influence with a nation to make then join your sphere.
As of February 10, 2014 this feature has been defined as Optional.