"do" carries a feeling of more confidence and certainty than "try"

"I will try my best" means that someone is putting the effort in to do that, but it doesn't mean that it is going to be the best. "I will do my best" means that they are putting a lot of effort in so they will do their best. They are doing the best that they can.

They are very similar and often used interchangeably. However, there can be a small difference in meaning. As Judy Anderson said, "do" sounds more certain, and also: "I will try my best" means that the person will make an effort, but not sure if the desired result is achievable. There is more attention to the uncertain result here. "I will do my best" is less about result and more about the effort.

I think they're the same.