The Handy Haversack Is Handier
This is a frame challenge. I disagree with the premise that a bag of holding is “obviously superior” to a handy haversack; and I disagree with the currently-accepted answer's interpretation that the difference in rarity is a historical hold-over, and that 5e made the main feature of the handy haversack immaterial through the description of the bag of holding. And I think that with that bit clarified, it will be easier to see why the handy haversack can be, well, handier.
What Does It Take to Retrieve an Item From a Normal Bag, Backpack, Chest, Etc?
Before considering magic items, let's make sure we're on the same page about some non-magical scenarios. There are no specific rules about interacting with containers like bags, backpacks, chests, and so on. Instead, these interactions and much more are covered by a general rule about interacting with objects.
Other Activity on Your Turn
…
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.
The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle. For instance, the DM could reasonably expect you to use an action to open a stuck door or turn a crank to lower a drawbridge.
— Basic Rules, Chapter 9: Combat, D&D Beyond
Later in the chapter, there are some examples of the sorts of things that could reasonably be done for free during movement or another action. I've included the ones that seemed relevant to this question.
Interacting with Objects Around You
Here are a few examples of the sorts of thing you can do in tandem with your movement and action:
- …
- withdraw a potion from your backpack
- pick up a dropped axe
- take a bauble from a table
- …
- fish a few coins from your belt pouch
- …
- take a book from a shelf you can reach
— ibid
Retrieving an item from a bag or backpack feels analogous to withdrawing a potion or fishing out some coins, so it could probably be done as part of another action. Similarly, picking an item up off the top of a pile or off the ground feels analogous to taking a bauble from a table or a book from a shelf.
But what about something larger? Would we feel comfortable letting a player use a single action for their character to reach into a barrel full of apples and retrieve a potion stashed at the bottom? Or to reach into a treasure chest full of gold and jewels to retrieve one particular necklace or gem mixed in with the rest?
For reference, a barrel is specified to have a capacity of 4 cubic feet, and a chest a capacity of 12 cubic feet or 300 pounds. A backpack or a sack (closest I could find to a bag) has a listed capacity of just 1 cubic foot. (Basic Rules, Chapter 5: Equipment, D&D Beyond)
I think many DMs (myself included) would be more inclined to call for one or more actions to search through these larger containers, and likely to call for a skill check if the character is under time pressure.
I think the key here is that the character is no longer interacting with a single object—one potion or one bauble—or even two, where the rules for object interaction say the first is free and the second requires a separate an action. They can't just jam their hand in and feel around—there are many objects between them and their goal, which must be sorted through, moved, and looked under. It sounds more like the Search action might be more applicable in this case.
Search
When you take the Search action, you devote your attention to finding something. Depending on the nature of your search, the DM might have you make a Wisdom (Perception) check or an Intelligence (Investigation) check.
— Basic Rules, Chapter 9: Combat, D&D Beyond
Note that the DM decides if a check is warranted and sets the DC, and can ultimately decide that the check is impossible given the circumstances and obstacles and choose not to call for a roll at all, or to resolve the search as multiple Search actions.
Taking things a step further, what about a character reaching through a 2-foot-wide opening into an unlit 4-foot by 4-foot by 4-foot room, filled with hundreds of pounds of gold, loot, potions, poisons, weapons, and possibly bodies, to retrieve a single potion in particular that they know is in there somewhere?
If it doesn't feel reasonable to assume a character could do that in a single uncontested action—if it feels like this might take several rounds or even minutes of searching, depending on the circumstances—hold onto that thought.
What Does It Take to Retrieve an Item From a Bag of Holding?
Both the bag of holding and handy haversack have similar rules in their descriptions around retrieving items.
Retrieving an item from the bag requires an action.
— Bag of Holding, D&D Beyond
Retrieving an item from the haversack requires you to use an action.
— Handy Haversack, D&D Beyond
I understand this as a specific exception to your general ability to retrieve an item from a bag or backpack for free as part of movement or another action.
Note that the rules still refer to these items as a “bag” and a “haversack”—they're still containers, just containers with a magically large capacity. Then retrieving an item from a bag of holding should follow the same rules for object interaction as any other bag—and it does, except that its description invokes one of the rules of object interaction that says
Some magic items and other special objects always require an action to use, as stated in their descriptions.
in order to deny the possibly of retrieving an item from a bag of holding (or handy haversack) for free as part of movement or another action. So retrieving an item requires at least an action.
That said, there's one big difference between a bag of holding and a normal bag: the inside of a bag of holding is a 4-foot deep, 64-cubic-foot space holding up to 500 pounds of crap with a 2-foot opening. Sound familiar? This is the exact same scenario as above, except that instead of a mundane room with a hole in the wall, the character is reaching into an extradimensional space with a hole connecting it to the Material plane.
There is nothing in the description of the bag of holding that makes this scenario any easier. The description does not say that you can use an action to retrieve an item, or that retrieving an item requires only an action—it just invokes the built-in exception to the object interaction rule to disallow retrieving an item for free.
Therefore, if we would not resolve the mundane scenario as a simple object interaction, it would not seem applicable here either—and like any other activity without specific rules, we fall back to improvising an action.
Improvising an Action
Your character can do things not covered by the actions in this section, such as breaking down doors, intimidating enemies, sensing weaknesses in magical defenses, or calling for a parley with a foe. The only limits to the actions you can attempt are your imagination and your character’s ability scores. See the descriptions of the ability scores in the Using Ability Scores section for inspiration as you improvise.
When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
—Basic Rules, Chapter 9: Combat, D&D Beyond
So, if you open up your bag of holding in the middle of combat, it's entirely up to your DM and your shared world-building what will be required to retrieve the item you want. Is the bag mostly empty, or did you explicitly say you left it on top of the pile in the the bag? Then maybe it would be a normal object interaction (but requiring an action). If not, maybe it's a Search action, or multiple Search actions. Is your character small? You may not even be able to reach the item you want to retrieve without climbing into the bag.
Again, to be clear, there are no specific rules for any of these things I've just mentioned, but there are no specific rules for most things characters want to do in D&D. This is why there's a DM, who can apply common sense to decide what's possible, what's achievable, and what it requires.
What About Retrieving an Item from a Handy Haversack?
The same reasoning about retrieving an item requiring at least an action applies to the handy haversack, but there's one important difference from the bag of holding.
When you reach into the haversack for a specific item, the item is always magically on top.
— Handy Haversack, D&D Beyond
Even if the capacity of a handy haversack is smaller than that of a bag of holding, it's still magically large. The main compartment has a capacity of 8 cubic feet, roughly the size of two barrels. But the item you want is always on top, with no searching required. This makes retrieval analogous to picking a bauble up off a table or a book off a shelf—it's right there, unobstructed and ready to be grabbed.
It's actually easier to retrieve the item at the bottom of a fully-loaded handy haversack than to retrieve pretty much any item from a normal backpack—you never have to feel around. If the item description didn't invoke the exception to free object interaction, retrieving an item from a handy haversack would almost certainly always be free. Since it does invoke the exception, retrieval just always costs an action: no more, no less.
A Bit More Comparison of the Descriptions of the Bag of Holding and the Handy Haversack
One other thing I'd point out is that the description of the handy haversack explicitly calls out that placing an item in the haversack follows the rules for object interaction.
Placing an object in the haversack follows the normal rules for interacting with objects. Retrieving an item from the haversack requires you to use an action. When you reach into the haversack for a specific item, the item is always magically on top.
— Handy Haversack, D&D Beyond
Compare this with the corresponding section of the description for the bag of holding.
Retrieving an item from the bag requires an action.
— Bag of Holding, D&D Beyond
Why did the description of the Handy Haversack feel the need to call out that placing an object in the haversack follows the normal rules? It's presumably not because of the subsequent sentence about retrieval requiring an action, because the bag of holding has pretty much the same sentence without the call-out.
So then it must be because of the sentence about the desired item magically being on top. This is different from the normal rules of object interaction when dealing with an 8-cubic-foot container—again, the equivalent of two barrels—where the desired item is normally buried under dozens of pounds of undesired bric-a-brac.
The description of the bag of holding felt no need to call out normal object interaction, but we know they must apply—otherwise there would be no way to put items into the bag. So then the normal rules of object interaction would also apply to retrieval (including the built-in exception to free object interaction, as described above)—and the normal boundaries of those rules would also apply, limiting what can and cannot be retrieved from a bag of holding in a single action the same way they do for every other container (or indeed small room).
Very Handy
This is (in my opinion) why the handy haversack has a higher rarity. Sure, it doesn't have the sheer volume of a bag of holding, but it lets you tote around up to 12 cubic feet or 120 pounds of whatever you want and retrieve anything from inside at a moment's notice. They two bags solve different problems: the bag of holding solves the problem of storage capacity, and the handy haversack solves the problem of immediate access to an arsenal of magic items (and also helps a bit with storage capacity).
It turns out there are plenty of ways to store large quantities of loot besides a bag of holding: rent a vault from your setting's Gringots analog; buy some wagons and horses to pull them; acquire a tower with a permanent teleporation circle for easy access; learn to cast Leomund's secret chest. Sure, the bag of holding is probably more convenient than all of these, but it's just degrees.
On the other hand, having repeatable, immediate access to any of 120-pounds worth of equipment is pretty powerful for a wealthy character. For example, fill your handy haversack with potions and wands. “Most potions consist of one ounce of liquid” (Dungeon Master's Guide, Chapter 7: Treasure, D&D Beyond)—an ounce of water weighs about an ounce, maybe double that to account for the glass, so 500 potions (62-ish pounds) probably fit in the haversack's combined capacity of 120 pounds. Wands have no listed weight, but are “about 15 inches long and crafted of metal, bone, or wood” (ibid), which would probably fit in even the smaller 2 cubic foot pouches of a handy haversack, and definitely in the main part.
Now you can always create any effect you want with at most one round of lead time—and if some of those potions are potions of speed, you can retrieve and use an item from your haversack each turn for one full minute.
Meanwhile the guy with the bag of holding is digging through a pile of gold, gems, and art objects for the potion of healing that he knows is in there somewhere.