Difference between "close up" and "close down"

There are many idiomatic usages where one word or the other is normally used, but the other would be understood to convey the same meaning. You normally close up a gap, or close down a mine, for instance.

You close up a shop, for example, at the end of every day's trading (then next morning you might open up again). If you close down the shop, the implication is the business is being "wound up", and will not re-open unless someone buys it off you and starts trading afresh.

Normally, close up a debate means to be the last speaker, summarising all that's been said on both sides of the issue. On the other hand, if you close down the debate, that means you (the chairman, for example) unilaterally end the debate by formally ruling out any further discussion.

TL;DR: Where there's a distinction, close down implies something more "final" than "close up".