Tie down is more emphatic than tie up, at least in most uses of the term. The physical and literal meanings have been well covered-- something that's tied up still has some leeway of motion, and if you want it immobilized you tie it down.
Both terms are commonly used in a figurative sense.
If you're tied up, you are delayed or hindered from doing something you were planning to do. Let's say you're late for a lunch date because the boss called you in unexpectedly with some extra work that had to be done right away. When you get to your lunch appointment, you explain the delay by saying, "sorry I'm late, I got tied up." The expression can simply mean you're busy, and is often used by secretaries-- "he's tied up at the moment, shall I have him call you back?"
If you're tied down, you are busy or occupied in a more long-term sense, and there are implications of obligation or commitment to whatever it is that's "tying you down." The phrase is often used when you break off a relationship because it's too confining, you want to be free of any commitment-- so you say "I just don't want to be tied down."
I don't really think the two terms are synonymous in the same sense that burn up and burn down are.
.
