Off the top of my head:
"Encourage" is used for people's actions/behavior.
Example:
"I tried to encourage Susie when she was feeling sad."
"Promote" is used for objects, things, plants, and functions.
Example:
I gave the plant a lot of food, water, and sunshine to promote better growth.
"promote" is also used to let someone know they are getting a higher paying job with more power/responsibility in their business.
Example:
"My boss promoted me to be a manager at Enterprise Rent-A-Car from being a cashier!"