Is there a difference in the meaning, between "appropinquare" + dative,
and "appropinquare" + ad + accusative?
Dictionaries are not very clear about it.
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In my view, there is a subtle difference in meaning. Crucially, note that the adjective propinquus, which expresses a state, selects a dative (e.g., in propinquis urbi montibus (Nep. Han. 5.1)). So the dative nominal associated to the verb appropinquare, which is derived from the adjective propinquus, is used to express the final static position attained by the subject: i.e., at the end of the motion event encoded in appropinquare, the nominative Figure subject is propinquus 'near to' the dative Ground object.
In contrast, the Prepositional Phrase formed by ad plus an accusative noun is related to dynamic motion + path (ad-) event associated with the verb appropinquare rather than to the inner stative adjective propinquus incorporated in this verb.
NB I: hopefully, my answer above can also be regarded as a sort of up-date of what I consider an interesting intuition found in this link.
NB II: alternatively, if you don't find the explanation above convincing, you could try to explore the connection between the dative of appropinquare and the so-called "directional dative": e.g., It clamor caelo (Virg. Aen. 5.451). The problem with this alternative is that this kind of directional dative is typically reduced to poetry. This construction is very rarely found in prose, where It clamor ad caelum is expected.
NB III: a second alternative would be to explore the well-known connection between prefixed verbs and datives. For a recent work and useful references therein, see also this paper. In my view, this second alternative can be claimed to apply to examples like ubi adequitavit portis (Liv. 22.42) but not to appropinquare urbi (Liv. 3.8.8). As pointed out above, the latter verb incorporates an adjective that already assigns dative. Furthermore, the difference between adequitare and appropinquare can be elucidated by the fact that the unprefixed variant of the former is incompatible with a dative (cf. the ill-formedness of *equitare portis), whereas the unprefixed variant of the latter is compatible with the dative (cf. the well-formedness of domui ignis propinquat (Tac. Ann. 15, 39)). The motion event of both appropinquare and propinquare incorporates the stative adjective propinquus, which is, as claimed above, the true assigner of the dative case.