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.
The rules as written say only that a bag of holding can be pierced from the inside. They don't say how hard a bag is to pierce. In other words, the rules don't actually answer this question explicitly... But that doesn't mean we can't work out the answer.
After all, we know that a bag of holding is a bag. More specifically, bags of holding are traditionally described as being sacks, though backpack-style bags of holding have also appeared in large numbers. Sacks are usually made of sackcloth, and backpacks are usually made of leather - and since we know what materials bags're usually made out of, we know how hard they are to cut: As per page 175 of the core rules, cloth has two HP per inch of thickness and 0 hardness, while leather has 5 HP per inch of thickness and 2 hardness; and as per page 459, magic items take damage as nonmagical items of the same sort unless otherwise noted: Since a bag is presumably less than an inch thick, it wouldn't be too hard to stick a knife through.
How likely is that to happen, though? If you place a knife carefully onto sackcloth instead of just tossing it in, the chances of it piercing the bag are almost zero, right?
Well... we know that a bag of holding has a maximum capacity measured in weight, and that it can carry up to that weight without trouble. A bag of holding is therefore unlikely to break just from the weight of stuff inside it. That implies that it's the cutting edge of an object, and not the amount of force behind it, that makes carrying a sharp object in a bag of holding dangerous. (We probably could have guessed that, but it's nice to have it confirmed.)
We also know that items place inside a bag can shift around. The rules for bags of holding don't actually say that their contents move when the bag is moved, but it's a common sense assumption for normal bags, and that means it's probably a safe assumption for magic bags as well. Even if your GM rules that the contents of a bag of holding don't shift around when the bag is closed, he or she will probably agree that they shift around when you stick your arm in to dig around for something - and that means that your hypothetical dagger could potentially be pushed into a position where it risks damaging the bag without your being aware of it. Uh-oh.
There's your conclusion: It's possible to put sharp objects in a bag of holding without damaging the bag... But there's risk involved.
But wait, you say, why is it so many people are adamant that you should never put a sharp object in a bag of holding? Well, a bag of holding is an expensive item that's used to store other expensive items; Many adventurers are understandably reluctant to risk losing what amounts to all their worldly wealth for a stupid reason. That, I suspect, is where the aversion to putting pointy things in a bag comes from.
Fortunately, even if your GM disagrees, it's not a real problem. Just buy a box, and put the box in your bag; Then, put sharp things into the box. It can even have rounded edges, if you like.
Best Answer
This will ultimately come down to how you interpret the phrase "each is like a bag of holding," from the description of Heward's handy haversack. If you think that that means the haversack inherits all properties from the bag of holding, except as described, then it's pretty clear what happens when you overfill the haversack. This is a reasonable interpretation, and in some ways the simplest.
If, however, you think (as I personally do) that the above line is simply defining the haversack as an extradimensional space, likening it to a bag of holding simply because that came earlier in the chapter, then it's worth taking a look at other extradimensional spaces in 3.5 and seeing if there's any precedent.
Unfortunately, each extradimensional space is different. Take the quiver of Ehlonna, for example. The quiver is limited by number of items and their type, not any sort of spatial requirements; it can hold eighteen javelins, but not a single rapier, even though that would almost certainly take up less space than the javelins. Or take the spell rope trick, which can hold exactly eight creatures, but size doesn't matter. It can fit eight titans but not nine halflings. When the rope trick ends, it just dumps the creatures out, rather than scattering them to be forever lost like a bag of holding. Rope trick also features the cryptic line
It doesn't elaborate. The only "hazardous" interaction in the rules is between a portable hole and a bag of holding (and, notably, there's nothing in bag of holding's description that prevents you from putting one bag inside another).
So, if it's not like a bag of holding, and it doesn't say explicitly what happens when it's overfull or breaks, how does a handy haversack work? Well, it's limited by volume and weight (it's actually worded as volume or weight, but that's a different argument), and pointiness doesn't matter. So if you fill the main portion of the haversack with exactly eight cubic feet of material, what happens when you try to stuff something else in?
Exactly the same thing that would happen if you were trying to stuff something into a normal space. If it's full of bolts of silk and you're trying to cram a sword in there, you'll ruin the silk. If you're trying to jam a glass jar into a haversack full of rocks, you'll probably break the jar. If you're trying to put a sword into a haversack full of swords, you run the risk of damaging the haversack, which has a hardness of 2 and 5 hit point per inch of thickness. D&D 3.5 doesn't have rules for splitting a bag at the seams, so some amount of DM discretion is necessary here, but it's worth noting that magical items still function just fine until they're destroyed. A suit of leather armor has 10 HP, so a leather haversack probably has a total of somewhere between 5 – 8 HP.
If the handy haversack were to be destroyed, the contents would neither be scattered across the cosmos nor deposited back on the material plane. They'd simply be in their extradimensional space, and you'd need to find a way to access it without the haversack. That's surprisingly tricky, short of sending a very tiny person there via plane shift or having already cast Drawmij's instant summons on the objects you want to retrieve. As discussed in this excellent answer, nondimensional/extradimensional spaces are demiplanes, which are difficult to access as per Manual of the Planes, Chapter Eight. Basically, you'll need to be at the location where the haversack broke to access it, regardless of what magic you use. High level options like crafting a cubic gate or just casting the spell gate will probably do it, or esoteric options like getting a 9th level planeshifter to swap a portion of the material plane with your broken haversack's demiplane. Basically, retrieving the contents of a broken haversack is a high level task, but the contents aren't completely lost.
If that sounds like a headache, it would be quite reasonable to rule that most extradimensional spaces from items dump their contents at the location of the item if it's destroyed, but that would be a house rule; RAW nothing happens, which means they're just stuck in an extradimensional space, and there's no easy way to retrieve them.