I need not devote too much space reiterating what you could find through a Google search of “Harvard Classics vs Great Books.”
If you want a set of books that would look impressive on any shelf, I don’t think you’d go wrong either way. But despite the admirably lofty goals of their editors, the collections do have some problems. I won’t try to cover all the reasons why these books spend more time on the shelf than not, but I’ll mention a couple.
When you do pull them from the shelf and open them up, they don’t provide the most pleasant aesthetic experience you’ve ever had with a book. The paper, the typeface, and the heft of the books aren’t necessarily inviting. To go cover-to-cover, end-to-end (with either set of books) would be a feat of endurance requiring a masochistic streak, for sure.
The choice of translations in these collections is a frequent source of criticism. While it might be convenient to have the various works in one collection, you can pick and choose better translations and/or more helpful commentaries elsewhere.
For me, at least, a little help makes all the difference. A while back, I tried to read the Bhagavad Gita, but didn’t get much out of it. Then, I acquired the edition of the Bhagavad Gita introduced and translated by Eknath Easwaran, who brought it to life for me.
Having aged out into the public domain, the Harvard Classics set is available online for free from numerous sources (and I think you can turn up some sites that offer downloads of Great Books). It isn’t hard to find hard copies of the Harvard Classics and Great Books of the Western World at used book stores.