I gather from the quoted part of post #17 that Mr Garner understands the two forms to carry the same meaning. He seems merely to be saying that one form has come to replace the other, which doesn't make sense if they have different meanings.
It may be that Garner holds that view; I agree that he gives the impression that he does. The difference in meaning that I see is rather subtle and only stands out in certain cases. There is more of an interior orientation with
BE accustomed to V-ing, in which construction
accustomed is an adjective. Its adjective status in that construction may be demonstrated by the fact that other copulative verbs besides
BE can be used there, and it can also be intensified by degree words:
They grew very accustomed to doing that.
In the
BE accustomed to [base V] construction, there is an exterior orientation. In this construction,
accustomed is a past participle, not an adjective; other copulative verbs besides
BE do not work; and
accustomed cannot be intensified by a degree word: *[strike]
They grew very accustomed to do that[/strike]. A sentence like
They are accustomed to do that may be analyzed as an adjectival passive. It would be a mistake to analyze it as a full verbal passive; it is
not equivalent to
Someone accustoms them to do that.
In most cases, I don't think it matters from a semantic standpoint which construction is used, so those of you who reject
BE accustomed to [base V] can proceed in your unimpeded use of
BE accustomed to V-ing instead. However, it is possible to imagine cases where there is a difference. For instance, I live in a condominium complex in which hired gardeners come to mow the lawns every Thursday. I don't know any of these gardeners; I don't know how comfortable they are with mowing the lawns here on those days.
Thus, if I chose to use
BE accustomed to in reference to these gardeners' practice of mowing the lawns at this condominium complex every Thursday, it would be more appropriate for me to use the adjectival passive construction, with its exterior orientation:
They are accustomed to mow the lawns here every Thursday. Who am I to say,
They are accustomed to mowing the lawns here every Thursday? I don't know what they feel accustomed to; I don't even know who they are, or even whether they're the same people each week.