The Bag of Holding (DMG pg.153) is an uncommon magic item and has a significantly greater volume than any of the pouches (or even all combined) of Heward's Handy Haversack (DMG pg.174), which is rare.
My confusion stems from the fact that, while the Bag of Holding is obviously superior to Heward's Handy Haversack, it's the more common item of the two. Why is that?
Some other info to consider:
In terms of fluff, the Bag is depicted as a simple satchel while the Haversack is depicted as a hiker's backpack with multiple straps and pouches. The Bag also has the disadvantage of looking similar to the Bag of Devouring. Regardless of contents, the Bag weighs 15 lbs. While the Haversack only weighs 5 (and I highly doubt 10 lbs. of difference is worth a difference of category in rarity, especially making the Haversack rarer when the bag holds approx. 8X more volume. To say nothing of weight.).
Other than that, they have the same rules regarding their extra-dimensional properties, interaction etc.
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Best Answer
This is a combination of history and what appears to be a small but significant oversight. The advantage of the Haversack over the Bag has always been that the Haversack always has what you're looking for on top, as compared to the Bag which is a disordered bag of stuff that's harder to sift through the more it holds (and it can hold so much).
The designers appear to have retained this distinction in the items' described function, and then made it immaterial during development by copy-pasting the standard 5e item-interaction boilerplate into their descriptions without making any adjustments to implement the functional difference they kept in the descriptions. (Easy enough to do, since when you're doing the rules polishing on the Bag you're not thinking about the Haversack, and it all looks fine, right?)
Traditionally (I mean back in AD&D, since in 3.x this is likely the kind of thing DMs would just handwave away), digging through a Bag for an item wasn't feasible during combat, so the discovery of a Haversack was a significant upgrade in, well, handiness. The capacity difference meant neither was strictly superior, but each had different pros and cons (the marginal utility of being usable in combat making the Haversack especially desirable, but still not strictly superior in every way). But none of this was nailed down in rigid action-economy terms then — there was no such thing as strict action economy terminology, it was just how the items' descriptions said they worked. The rarity difference in the 5e items appears to reflect a design intent to limit access to the especially-desirable item usable at combat speeds; later negated by failing to implement this small but significant distinction during the development stage.
So it just seems to be an oversight. RAW, the Haversack is the clear loser. For many DMs, the question ends there (plus a bit of head-scratching at the designers). However, 5e DMs aren't bound by RAW and are encouraged to make 'fluff' matter in their games, if they so desire.
If you want to emulate the traditional utility of the Haversack, make Bags of Holding able to be interacted with using an action... but it takes multiple before the user finds what they're looking for. You can either nail this down (at which point you're into making house rules to taste to determine exact number of actions), or just say that it's only possible during non-combat time and handwave the exact time it takes. No matter how you implement it exactly, this will leave the Haversack as the true champion of handiness that's usable with a single action to get exactly what you want.