Bør is more in the sense of "I should do this and I might" and burde more in the sense of "I should've done this but chances are I won't or it's too late", though both will be user interchangingly with no real difference.
One way to think about "burde" is in the sense of "burden". It seems to indicate something one is obliged to do or to bear.
In that case, wouldn't a better translation be "I have to go to bed"?
We'd probably use "I should go to bed" in the sense that it's late, and I won't get enough sleep if I don't go now, but I probably won't go just yet.
In that case, I think it means a child have to go to bed... Or someone before a hard day at work...
Burde is some sort of moral obligation.