What I am seeking here is the logic that makes the Form of the Beast so enticing. From what I understand, it would take a 2d6 greatsword attack and reduce it to a 1d8 bite/tail attack, or two separate 1d6 claw attacks, which seem weaker to me. Are the additional features that strong? Is there something I missed in making them simple natural weapons? Is there a way to make it more powerful than the greatsword? What am I missing?
I'm currently playing a Beast Barbarian. Here's what I've noticed.
Claws are especially good because you get N+1 attacks. Others have addressed the damage increase, but I think your question misses an important fact: the bulk of your damage comes from the Rage+Strength bonus, not the dice. Also, if you have advantage, you're have it for two attacks instead of just one - and you can give yourself advantage from Reckless Attack!
The downside of spreading the damage into multiple attacks is if you're fighting enemies with damage reduction... but that's pretty rare in DnD 5e. Compare that with forcing more CON saves for spellcasters with concentration spells, I think more attacks comes out ahead.
The tail is nice for playing defensively, as others have pointed out. Since it has Reach option also gives you a bigger radius from which to make Opportunity Attacks.
I don't use the Bite form very often since I rarely enter combat below half health. Still, if your party is running low on healing, this is a nice option to stretch your HP.
And now my favorite benefit: the versatility. You have all of three options (Claws, Tail, Bite) available to you before combat even starts! You don't have to choose a feature at the end of a rest; you choose it once you start raging. I love playing casters and half-casters; having the flexibility to adapt my tactics based on the encounter is very appealing.
You're also not dependent on your weapons. It's not every day that adventurers needs to fight without their equipment, but if you do lose your equipment - whether through imprisonment, shipwreck, theft, or confiscation - the Beast Barbarian is one of the best classes to keep going.
You'll probably give up the Great Weapon Master feat. There's not a lot of benefit to two-handed weapons. On the plus side, you get to have a shield. And it wasn't as though you have GWM and are stuck with it forever; now you can pick a different feat like Sentinel, Charger, Resilient, Fighting Initiate, Mage Slayer, Toughness, Shield Master, Slasher, or whatever else catches your eye.
You're only looking at the raw dice.
The Bite is capable of restoring hit points. Especially for a raging barbarian with damage reduction, gaining some extra hit points every round can go a long way. (If you're not hurt, don't manifest the bite form, it indeed won't be better than your sword.)
The Claws give you two separate attacks, which means you can cut down two weaker foes in one go, increase your odds of landing at least 1 hit, and if you hit both you deal your Strength damage and Rage bonus damage twice, so it can outpace the Greatsword in damage.
The Tail gives you Reach, which means you can attack enemies from more than 5ft away. It also gives you a defensive tail-sweep to reduce the chance of being hit by nasty attacks.
Each of these can be the right choice in some encounters, giving you extra defenses, extra attacks, or healing. But you'll see that when using it in play; if you're just looking at "how many dice do I roll", they all seem lackluster.
Before we consider the benefits of each attack, lets make a Very Standard Barbarian, we will call Vicky Stone. We will play with the standard array, putting our racial bonus into CON and STR to bring these scores up to 16. Vicky took the average rolls for health on level up, so at 3rd level Vicky has 35 hit points. Vicky can have an AC of 14 with Unarmored Defense, but 15 with scale mail and putting 12/13 in DEX, which I think by 3rd level Vicky will have come across enough money to purchase.
For the sake of simplifying the problem, we are only going to look at one monster Vicky is Fighting: a CR 3 creature, with the quick monster stats from the Dungeon Master's Guide p. 274
Lets consider 4 of Vicky's options for combatting this creature: a greatsword, a Bite attack, dual Claws, or the Tail (with 2 choices of source of damage). (we will ignore critical hits for simplicity).
The Creature has a +4 to hit, so it will be hitting Vicky's 15 AC 50% of the time. Dealing on average 22 damage/hit, halved for a barbarian's rage, the Creature will average out to 5.5 DPR (damage per round) on Vicky. This means Vicky can be expected to be knocked unconscious on her 7th round in combat.
With her relatively low DEX, she likely is going after the Creature in initiative, so she will be able to attack 6 times. Her greatsword deals 2d6+5 damage on a hit and is hitting 65% of the time against the Creature's 13 AC (averaging 6.5 DPR), so we can expect Vicky to deal 47 damage before being KO'd by the Creature. Not too bad, but let's see how it stacks up to the other options.
With this attack, Vicky is able to wield a shield while in her free hand. This means the Creature has only a 40% chance to hit this time. So the creature will now be dealing 4.4 DPR. Vicky is also healing 2 hit points 65% of the time once below half health, so after taking 18 or more points of damage (4 rounds), the effective DPR is reduced to 3.1 DPR; giving Vicky an additional 6 rounds conscious. With 9 rounds to attack with a 1d8+5 weapon, still hitting 65% of the time, Vicky can on average deal 55 damage before being KO'd.
Maybe not intentional, but the RAW interpretation of these claws allows the player to still wield a shield while using the second attack. So lets just take that 4.4 DPR - same as the bite - and see that Vicky will be knocked out by the Creature's 8th round of attacks.
So in 7 rounds Vicky makes 14 attacks with her claws, dealing 1d6+5 damage 65% of the time, for a total of 77 damage before being KO'd.
Tail; with Tail attack
Oh boy is this going to be a drawn out one for Vicky here. Her shield, plus her 1d8 bonus AC tail reaction, bring her effective AC from 15 to ~21. So our Creature now only has a rounded 20% chance to hit. That 22 DPR has been reduced to an effective 2.2 DPR, so Vicky can survive this onslaught for 16 rounds.
With the Tail attack 15 times dealing 1d8+5 damage/hit, Vicky will be able to dish out 92 damage before being KO'd, which may be enough to defeat this creature before it defeats Vicky.
Tail; with Greatsword
Okay, so you read all this and still only care about swinging the big sword. The massive potential boost to AC from the tail still makes this a very effective subclass, even if that is the only benefit you want. ~4 added to AC as a reaction means the Creature is now hitting 30% of the time, for an effective 3.3 DPR; Vicky can expect to be conscious for 11 rounds.
This will give her 10 rounds to strike with the greatsword now, bringing her to 78 damage before being KO'd.
The Moral of the Story
Obviously, there is another several members of the party to consider, other creatures in the fight, multiattack, legendary actions, aggro, other ACs and damage rolls, bad luck, good luck, etc. It is also extremely unlikely your DM is going to ask you to enter a one-on-one slug fest for 16 rounds against a creature. However the effects of these beastly weapons are evident, and can be just as effective over numerous encounters and multiple enemies, as long as Vicky can enter a rage.
Also, while the tail seemed the most effective for total amount of damage dealt, this is very situational to the scenario set up here. More enemies decreases the tail's effectiveness, as Vicky can only use a reaction once per round; however the bite could mean your healer can focus on other targets, the claws can hit an additional creature at each round, and so on.
But even if we just look at the flat DPR of the Greatsword when compared to the Claws, when adjusted for the 65% chance to hit the Greatsword is dealing 7.8 DPR, and the claws are dishing out 11.05 DPR.
A greatsword takes two hands to wield. The claws require only one free hand. A bite or tail attack requires neither hand. Swapping to these options gives you more free hands to use for things other than your weapon. Several of the other answers have already identified the obvious advantage to this: you can use a shield. There are a lot of assorted other things one can do with a free hand, but the other big one is grappling.
If you grapple an opponent and shove them prone (both on PHB pg. 195) they can't stand back up. That gives you and your melee allies advantage on attacks against them, and them disadvantage on attacks against you and your allies. It also saves the perennial problem of how, as the party member who'd prefer to be taking all the hits, to prevent enemies running off and whacking your wizard. That makes it an excellent crowd control option for martial characters, even before considering the (oft hilarious) benefits of being able to drag your foes all around the battlefield. As a barbarian, Rage gives you advantage on all your Strength checks, making grappling an even more reliable option for you.
But grappling requires a free hand, per enemy you wish to grapple. So you can't use it at all while wielding a greatsword. If you're grappling one enemy, you need to choose either a shield or a weapon. If you're grappling two enemies at once, you get neither. That leaves you with simply unarmed strike damage, which even with the Tavern Brawler feat is just a d4. Unless you have a tail, or a bite attack, in which case you can merrily hold two enemies (or an enemy and a shield), while still using a d8 weapon. This makes the Path of the Beast an excellent option if you plan to do much in the way of grappling.
I'll look at it at level 5 with 18 strength while in rage.
The baseline is a 2 greatsword swings for 4d6+12 or 26 damage.
The second baseline is a shield and 1d8 melee weapon for +2 AC and 2d8+12 or 21 damage.
With a Bite, you can do a greatsword attack and bite. You deal 2d6+1d8+8 (19.5) damage and heal 3 HP.
We'll go claw and shield for 3d6+18 (28.5) damage.
Arguably using item juggling you can even pull off 4d6+20 damage (two short swords, each turn either sheath or draw one of them, to activate two weapon fighting bonus action attack) for 34 average damage.
Or, make a greatsword attack, let go with one hand and make 2 claw attacks, for 4d6+18 damage (32).
This one offensively doesn't boost you, but it adds +1d8 AC as a reaction every turn, which can be pretty beefy. It is the best non-two-handed/heavy reach weapon in the game; so you can use it with a shield.
The greatsword damage is 26. Claw can do 32-34, significantly more. Or it can use a shield and still exceed greatsword damage at 28.5!
Other types of barbarian
Now, this doesn't guarantee it matches other barbarian types. But the point is that it a benefit, not that it is optimal.
Zealot with greatsword is doing 5d6+16 (33.5), and has boosted accuracy on 5.5 of it.
You're missing a few things.
The general evolution goes like this.
"2d6 > 1d6, so greatsword is better than claws/bite/tail especially once you get to level 5" -> therefore, beast barb bad
"wait. 1d6 or 2d6 doesn't matter much if i'm adding +4 from str and +2 from rage to each attack... even magic weapons don't close this gap fast enough to matter if it's 2 attacks instead of 1 or 3 attacks instead of 2." -> therefore, beast barb great!
"more attacks is good, but great weapon master exists and barb gets automatic advantage if they want it. +10 > +6, especially if it's +20 (two attacks) vs 6 (still that same 1 extra attack)" -> therefore, beast barb bad
"wait. if I have polearm master and great weapon master, I can attack with the haft as a bonus action... meaning I get 3 attacks anyway." -> therefore, beast barb bad
Beast barb's only really good for Small sized barbarians. That said, being Small and a Barbarian is kinda ass optimizationwise so it's still not a super great thing.
The tail has some utility regardless but only some. Generally, other kinds of barbarian (ancestral, bear totem) are superior for nearly any purpose.