Since only seven questions in total are currently tagged spam, three of which are also tagged spam-handling, and spam-handling is the tag we've pretty much all been using lately, you might be wondering why I'm even asking this, rather than just retagging them all spam-handling.
I've heard that's what other child metas use for this.* I guess a SEDE query could show this rigorously, but a quick look through the top sites' child metas seems to confirm it. The network-wide meta site also uses spam and not spam-handling.
There hasn't been a difference between spam and spam-handling. But someone may ask a question about the promotions policy, along the lines of "Would it be spam if I...?" There, the topic of spam applies, but the question is not really about spam-handling. I don't think we want the granularity of separate tags, and spam is the slightly broader one, so I think we should use that.
So long as we make one of the two tags a synonym of the other, I think the outcome is fine. But I think it would be best if we used spam as the main tag.
One benefit of making them synonyms instead of removing one of the tags from all the questions that have it is that we can avoid bumping a whole bunch of meta questions by editing (seven to detag spam, and far more to detag spam-handling). Bumping multiple meta questions has quite a visible effect since meta is slow, though it's not actually harmful to do so.