Plan to do means to plan or intend to do something, indicating that there is no specific action, just an idea or plan. I plan to go to the gym this afternoon. )
Be to do means that you have decided to do something, which means that you have an action plan and are about to do it. I'm going to meet my friend later. )
Plan to do is usually used in more formal and commercial occasions, emphasizing the arrangement and preparation of the plan. Be to do is more commonly used in oral communication, emphasizing decision and will.
When expressing intentions, the tone of the plan is euphemistic and vague, which may be changed or cancelled; Be going to do is more firm and clear, indicating that the plan will be carried out.
In a word, what is planned and will be done expresses the future intention, but what is planned and future is emphasized, while what is to be done emphasizes action and decisiveness.