I'm not an economist either, so I may be getting this wrong, but: the national debt is how much the country owes, full stop; the budget deficit is how much more the country intends spending than what it's bringing in. So the one leads to the other; but you can balance the budget or even be looking at a budget surplus, and still be carrying a debt, see? (So they tend to differ by orders of magnitude; on this side, our debt is some low-ish number of trillions of US dollars, but the deficit each year is some middling number of hundreds of billions.)

Put even more simply: running a deficit makes it difficult or impossible to pay down one's debt.