Why tanks feel bad to play/feel weak
The Issue with Jungle Tanks A lot of ADCs are taking flat armour penetration builds. This should benefit tanks, since tanks will stack armour and be affected less by the armour penetration that the ADCs are building.
However, jungle tanks will typically go Cinderhulk as their first item, which offers no armour and are on a lower income than the ADC is. As a result, it is not uncommon for the jungler to not have a completed armour item before the enemy ADC has their first 2 armour pen items. This is compounded by the issue that many tank junglers have slower jungle clear, as their focus does not lie in damage but in crowd-control and survivability.
Tank itemisation from the jungle SUCKS early to mid-game. Whilst building health grants you effective HP against both AP and AD threats, your second item almost always feels bad to purchase. Against an armour-pen ADC you have the choice of Randuin's Omen/Dead Man's Plate, at which point you're probably sitting at a little over 100 armour. At this value, flat armour pen is still very effective. An ADC with Youmuu's Ghostblade will have more effective armour penetration than one with a Last Whisper item and one with Youmuu's plus Black Cleaver will be extremely effective against you.
And, having built this defensive item that already feels bad because it isn't doing sod all for you, in terms of stat priority for many tanks it blows. The majority of tanks want to prioritise cool down reduction to bring their ult cool downs as low as possible as well as to allow them to use their low-cool down crowd-control abilities multiple times in a fight. Champions like Sejuani and Amumu cannot afford to go for Randuin's Omen or Dead Man's Plate as a second item.
So where does this leave us? If you have to itemize for CDR AND armour your options are Frozen Heart and Iceborn Gauntlet. These options do not provide any health and leave you very vulnerable to an AP source of damage. The, at this point, 350 HP that you are receiving from Cinderhulk is not going to save you and it is not unlikely that against an AP threat you will be one of the squishiest members of your team as a tank. I don't think I need to explain why this is a problem. Additionally, these items synergise very poorly with the bonus HP scaling of Cinderhulk, as well as the HP scalings on some jungle tanks such as Sejuani which makes them feel bad.
And all of this is under the assumption that you're even free to itemise armour. If they have a fed AP threat you will be expected to buy Locket of the Iron Solari (which has its own issues), this leaves you totally vulnerable to the enemy ADC and is not an item that any jungler ever wants to build especially as a second item. Even in the case that you aren't expected by your team to build it. Spirit Visage is honestly one of the best tank items in the game, offering a perfect stat profile for tanks, a useful passive to many tanks (and you know that itemisation sucks when the tanks, like Sejuani, who do not benefit at all from it are buying it) and has synergy with Cinderhulk as it has that HP to gel with the HP scaling of Cinderhulk.
The result of all of this crap is that 2-item tanks feel bad. 1 item tanks aren't that strong because Cinderhulk is not a strong item early game (with a similar cost to Sunfire Cape and only a fraction of the stats, yet required because jungle items...). Many jungle tanks don't even build Cinderhulk because of this. Last I checked, Amumu doesn't, Zac doesn't, Gragas doesn't. Sejuani is locked into it due to scaling off of HP, but can build Runic viably since she has better AP scalings than Amumu does. By the time you have 3 items has a tank - probably Cinderhulk, the Frozen Heart you were forced to buy because you needed CDR, but prevented you from using your cool downs because the enemy mid laner was able to one-shot you because you have no magic resist, and a Spirit Visage with a potentially useless passive on you because you needed HP, CDR and Magic resist so badly that the stats were too good to pass up.
And now? Your enemy has a carry top lane with either Black Cleaver or innate ability to duel tanks, or both (cough Fiora cough) to shred your armour, the ADC has a load of flat armour penetration and also a black cleaver. Your 150 armour becomes around 100 and we're back to them having a last whisper equivalent that also happens to be obnoxiously effective against squishies.
OR you've stacked health, their flat armour pen and black cleaver combination is even MORE efficient, since y'know, you have less armour so the flat pen does more but you have no CDR and your back line just dies because you can't peel for them as effectively. Oh and you still die within 2 seconds but it's something, right?
But don't worry, I can keep stacking armour, right? It's not like damage out-scales defence to give tanks a crappy end-game, is it? ...oh wait.
Honestly, I feel like the problem with jungle tanks lies with Cinderhulk itself. The immolate passive is exactly what jungle tanks need but the stat profile on it is kind of ass. Additionally, the complete lack of mana regeneration on it blows for mana-gated tank junglers. Mana regen, equal to a proportion of immolate damage done to monsters I think would be appropriate. That's a whole other topic though.
TL;DR: Mid-game itemisation feels bad due to flat-pen builds from ADCs, who eventually build LW items and/or crit items and outscale the tank.
Top Lane Tanks Top lane tanks have more availability to itemise effectively, as long as they're not against an AP top, a splitpusher they need to be able to duel, have a core item that doesn't offer one or both resistances such as Rod of Ages or require some form of mana sustain.
Against an AD top you have a much easier time and things feel a lot better, in my opinion. You can start the game with a Sunfire Cape, and if you need CDR you can itemise for that with component items. Kindlegem and Glacial Shroud offer enough to sit on early game, and build into items that you're probably going to be building anyway. With less HP scaling, Frozen Heart/Iceborn Gauntlet feels a lot better to purchase and you are simultaneously itemising against the AD carry. If the enemy AP is ahead you can opt to build multiple HP/Armour items and still have effective HP against the AP threat, and Spectre's Cowl and Negatron Cloak give you cheap options to offer some reactive defence against an AP who picks up a random triple.
The issue comes when you fulfil literally any one (or more!) of the above requirements.
Rod of Ages feels bad to purchase as a tank and I don't think any tanks are building it anymore. Tanks lack viable mana options outside of Frozen Heart/Iceborn Gauntlet (notice how these 2 items keep coming up as being compulsory items) and if they're against against AP they have to suck it up or opt for RoA. If you're an AD tank? Good luck!
Split-pushers are typically AD and therefore don't always offer the same challenges to itemising against AD carries. They can lock you into building certain items, but again, these are usually effective against ADCs. Honestly, anti-splitpush itemisation is one of the rare occasions where tank itemisation doesn't feel bad even though it feels very forced. Singed can feel bad to itemise against but he isn't currently meta-enough to be a significant issue.
And then we have itemising against AP laners. Honestly, this feels bad for everyone. Spectre's Cowl as a laning item, when combined with innate sustain, still feels overtuned. Issue is, that is the saving grace for itemising against AP. Itemising Spectre's Cowl/Bami's Cinder gives you effective stats against your laning opponent but allows you to easily build Sunfire for teamfighting.
As a whole, I don't believe that top lane tanks are in anywhere near as bad of a spot against ADC relative to jungle tanks. Mana itemisation for AD and against AP sucks. Not an issue with flat pen ADC since you CAN build 2 armour items against them the majority of the time.
The issue with top lane tanks, however, is that playing a top-lane tank can feel like handing your opponent a free laning phase win against you if they pick any of the popular carry top-laners, which excel at killing tanks. Fiora, Yasuo - particularly bruiser Yasuo - etc. etc. Tanks, due to their high base damages and crowd control should be lane dominant. Whilst their kits will be inherently poorer for laning, they should have more damage AND survivability early game than a carry-based top laner, who will have lower base damages but higher scalings.
The result of tanks having poor laning phases is that they will hit the mid-game behind. If they're 800 gold down on their laning opponent that can mean the difference between a full item and components. Just 1 kill can be enough to create this kind of discrepancy, with the 300 gold that the laner gets from the kill, the CS they deny and the pressure that they get in lane from that point on-wards.
This results in them having a poor mid game. Sound familiar? They get blown up mid-game, the enemy carries get more gold than they do, they continue to get blown up.
Tanks, both top lane and jungle who get ahead can be EXTREMELY effective, and are a blessing to their team. Having the extra half-item can make a tank's stat profile a lot smoother, with Kindlegem or Glacial Shroud providing CDR, Spectre's Cowl or even Negatron Cloak allows them to deal with AP threats hugely more effectively to compensate if they itemised Frozen Heart. The issue is that they have to be ahead to feel effective, and that's the opposite of how a tank should feel.
TL;DR: Top lane meta, ADC itemisation and jungle itemisation results in poor 2-3 item teamfighting for tanks. This results in them never feeling strong as the damage will outscale them late-game and they receive less income, becoming further and further behind. Tanks are required to be ahead to feel effective.