As far as I know, there is no difference in meaning between modal-verbs and modals, since mood in English always refers to a verbal category [Edit: I was wrong; F.E. has pointed out the application of "modal" to adjectives, adverbs and nouns in English grammar]. I think modals should be a synonym of modal-verbs, but I don't have enough rep in the tag to suggest a synonym on my own.
Why "modal-verbs" should be the main tag, and "modals" the synonym
There are no followers for either tag, so we don't have to worry about disrupting distinct communities. Since modal-verbs has many more questions (283), it seems like it should be the main tag. However, modals does have 23 questions, enough that I'm not comfortable deleting it from questions one-by-one, and also I think enough to show that it is a tag that is likely to be used again in the future if we don't make it a synonym.
It is true that some linguists refer to modal adjectives, adverbs and nouns. However, I haven't seen any questions about these on ELU, and I just looked through the entirety of the modals tag.
Side-question: what should we with the tag "semi-modals"?
Since only five questions were tagged with semi-modals, all of them were also tagged with modal-verbs and/or modals, and the tag has no followers, I just went ahead and deleted this tag from all questions.
Just because the line between core modals and non-core modals might be hazy, that doesn't mean that the label "semi-modal" is not useful. The class of core modals (or central modals, or whatever term your grammar uses) is in general mostly consistent no matter which grammar framework you use. The non-core modals are on a gradient, and often have different classifications w.r.t. the various grammars out there. Non-core modals--such as "BE going to V", "had better", "dare", "need", etc.--shouldn't be grouped with the core modals (e.g. WILL, CAN, etc.).
- It is not used much; currently it only has 5 questions
- It is generally always accompanied by the tag modal-verbs and/or modals; this makes the tags on a question more cluttered
- the above is also evidence that regardless of what is technically correct, in practice question-writers already do group non-core modals with core modals
- the people interested in questions about semi-modals, both from an asking and an answering perspective, are likely to also be interested in questions about core modal verbs
- Different sources have different definitions of the dividing line between true modals and semi-modals; not all sources use the classification of "semi-modal." (I don't have any direct examples right now, but there is some discussion of this on the Wikipedia article about English auxiliaries that has citations said to support this.)
Alternate proposals of mine
If the community decides to keep the tag "modal-verbs" restricted to the core modal auxiliaries, as its tag wiki suggests, it seems to me it might be more clearly named modal-auxiliaries or something like that.
The tag wiki for modal-verbs should probably also mention modals as an overarching category tag and semi-modals as a tag for other non-auxiliary verbs or phrases that are similar to modal verbs. These two tags also need tag wikis written for them that describe their scope.
There are also a few incorrectly tagged questions that we would need to retag. (Some questions currently tagged "modal-verbs" deal exclusively with semi-modal expressions.)
I guess I would also like a policy on whether double-tagging questions with both modal and modal-verbs, or both modals and semi-modals, is encouraged, discouraged, or up to the preferences of the individual tagger. If double-tagging is discouraged, I don't even know which questions would use the tag modals, as the more specific tags modal-verbs and semi-modals completely cover all the questions to date. I don't think double-tagging should be encouraged because, as I said earlier, it clutters up the tags beneath questions, and it is also hard to do consistently. And if it's up to each individual tagger (the usual de facto case even if there is a policy) then there is certainly going to be inconsistency.
A similar issue: there are a number of questions that deal with both core modal auxiliaries and semi-modals. How should these be tagged: a) with only modals b) with modals, modal-verbs and semi-modals c) with modal-verbs and semi-modals.
In sum, if we keep all three tags, I'd propose changing their names to modals, modal-auxiliaries and semi-modals, and coming up with a clearer policy of where each should be used and how they differ from one another.