The basic difference between these two things is that a budget deficit is only about one year while the national debt is cumulative.
The budget deficit is the difference between the amount the government receives in revenues in a given year and the amount that it spends in that same year. When the government spends more than it gets, it has a budget deficit.
When the government does this over a period of years, the deficits build up. When you add the deficits from each year (plus interest and minus any payments that have been made) you get the national debt.
So the debt is, essentially, the cumulative total of the deficits from all the previous years.
a budget deficit is the amount by which the government's income fall short of the expectations set forth in its budget over a given period of time. whereas, national debt is the sum of what the federal government owes its creditors.