Difference between sunk cost fallacy and an appeal to closure?

Closure is different from sunk cost. With closure, it is simply claiming some truth value out of exhaustion or a desire to be done with the topic. With the sunk cost, it is the desire to keep putting effort into something (and/or claiming a truth value) to avoid the harsh truth that all previous effort have been futile. So I would wouldn't really classify that one as "just finishing something off." The real motivation is to avoid taking the loss.

They have similarities but the reasoning is different

The appeal to closure is effectively a way of saying, "X is true for the sake of providing closure to the issue." In other words, you're emotionally exhausted and want to put an end to the matter, so you accept something which might be false, without examining it first. This is a rationale for stopping something.

The sunk-cost fallacy is more of a cognitive bias where a person says, "I already put resources into X, which would be wasted if I give up - therefore, I should put more resources into continuing X." In other words, since you've started, you might as well continue, since you don't want your previous effort to be in vain. This is a rationale for continuing something.

See the difference?