They are very similar; often we could use them interchangeably- "I am to get married next year!" "I will get married next year!"
However, "I am to do..." feels a little old-fashioned and isn't as commonly used in my experience. It also could suggest an obligation- what you are SUPPOSED to do or what has been planned for you- while "I will do..." suggests more what you INTEND to do or what you plan for yourself. The difference is very subtle.
A: What part did you get in the play?
B: I am to be the king's servant.
A: Oh, that's not what you wanted. Do you still want to be in the production?
B: I'm not sure. I will think about it.
When in doubt, use "I will do." It's a little more common. Hope that helps!