If you have gold/diamond status HGI is the way to go since you get a cooked to order breakfast and the items are quite good (French toast, waffles etc)
Hampton Inn blows away everyone else in the budget segment though for its breakfast (was at a HIE last week, ugh)
Between the 2 brands though, it’s almost a property by property discussion. I’ve stayed at some phenomenal HGIs (Singapore Seragoon) and some fantastic Hampton Inns (big shout out to the Bend, OR location). With the older properties though, both of them can drop the ball. Look at photos and the guest reviews on Google and you'll have a good feeling on which one is going to be better for you.
Also if you’re like me, you’ll often skip breakfast so it’s not as big of a consideration.
Every chain has this though: The new Four Points by Sheratons are fantastic but the old ones really suck