With regard to to your question
Does the meaning of a sentence ever change between these two alternatives of ', which' and 'that' or is this just a style choice?
there is a famous instance from the 1984 Republican Party platform drafting debates that illustrates the significance that the choice between the two forms can have on sentence meaning. A draft of the taxation plank of that platform included a sentence that initially had this wording:
We therefore oppose any attempts to increase taxes which would harm the recovery and reverse the trend to restoring control of the economy to individual Americans.
Observers noted that if which were read as that—that is, as a restrictive clause—the effect would be that the platform opposed harmful tax increases but held out the possibility that certain tax increases might be harmless or even beneficial. (This was 1984, I hasten to reiterate, when such a reading of a Republican-authored document would have seemed not at all absurd.)
On the other hand, if which were read as the first word of a nonrestrictive clause, the wording "which would harm the recovery and reverse the trend to restoring control of the economy to individual Americans" would amount to a characterization of any tax increase.
According to Steven Hayward, The Age of Reagan: The Conservative Counterrevolution: 1980–1989 (2009), a congressman from Texas named Tom Loeffler noticed the ambiguity and insisted that the sentence be clarified by adding a comma before which, thereby confirming the nonrestrictive nature of the phrase "which would harm the recovery and reverse the trend to restoring control of the economy to individual Americans." Loeffler's effort succeeded, the comma was inserted, and the Republican Party began its tradition of explicit theoretical opposition to all tax increases.
With regard to your question
Is Word always correct in suggesting either a comma before 'which' or else using 'that'?
the notion that a word-processing program might be correct in trying to force users to make an either/or decision when a third legitimate choice (restrictive use of which) is available is, in my opinion, untenable. I have used that and which in their Word-approved ways for so long that I do it almost unconsciously—which (I hope) is fine. But many writers, both in the past and in the present, use which in place of that in certain situations, and I have no trouble understanding what they mean. I see no reason to let Word dictate correctness or acceptability in such a case.
For me as a reader, a problem arises only when (as in the Republican Party platform example above) a writer perhaps intentionally uses which ambiguously to avoid getting pinned down to a particular meaning. A comma is an easy thing for a reader (or writer) to miss, after all. But it seems to me that Word's effort to make writers choose between 'that' and ', which' may lead to more misstatements (from writers who mean one thing but haplessly choose the other under pressure from Word) than would have resulted if the program had simply butted out and let writers express themselves naturally.