Everyone knows what a mission statement is. It's a really long paragraph with all kinds of vague words that you wrote when you were applying for a loan to start your business. Truth is, you could probably swap your mission statement with any other business out there and it would suffice. A brand promise, however, is more tailored to your consumers. It's a one sentence summation of the mission statement focusing on your key differentiators.
Key point: the brand promise is different from a tagline. Who knew there were so many statements?! A brand promise is probably still a little lack luster as far as generating excitement goes. However, it is used as a good point of reference for all future endeavors. If, for example, you brand promise is "to provide great tasting food at a low cost" then you probably wouldn't offer $40 hamburgers to your patrons.
At the end of the day, the brand promise is used to sum up your mission statement and the tagline is used to drive action. They're both related, but they serve two distinct purposes.