Modern tanks are pressurized, so they can deal with the ominous triumvirate of nuclear/biological/chemical weaponry. Being inside one that takes a completely non-penetrating hit is probably not really a big deal. These tanks are on the order of 60+ tons (the Tiger was somewhere along those lines as well, although most WW2 tanks were much lighter) so a hit isn’t really going to bother it. I’ll leave it to someone else to hunt up the shell mass, muzzle velocities, and do the comparative math to see how much the tank would be affected, but basically it’s functionally nil – you just don’t see weapons strong enough to do that (with one caveat, which I mention later)
Imagine being inside a steel drum (albeit not directly connected to the outside, seats and such) that gets banged with a sledgehammer. It’d ring your bell, but probably not be any real concern physically. Psychologically might be a different story 
In general, if you live long enough to realize that your tank just got hit, you’re fine.
As for tank weaponry, the main modern anti-tank weapon employed by tanks is the kinetic dart, the current US piece being the M829A2 Armor-Piercing Fin-Stabilized Discarding Sabot. This is a (I’m guessing) 50mm wide, 1-meter long depleted uranium dart carried inside a spool-like sabot to fit it in the 120mm smoothbore (German ) gun tube on the M1A1/A2/A2SEP main battle tank. There’s no explosive at all in the round itself, it’s merely a giant spear of dense material fired at high velocity that pokes through the armor and sends high-speed fragments (of the armor, and the penetrator itself) inside the crew compartment, turning the crew into jelly, starting fires, possibly detonating ammo.
Another modern AT round is HESH, High-Explosive Squash-Head – this is what Whack-A-Mole was describing. It basically fires a large slug that flattens out against the enemy’s armor upon impact, giving it a good whang. Basically a king-size sledgehammer for your steel drum 
The idea is to flake off fragments of the armor inside the crew compartment, turning the crew into jelly, starting fires, possibly detonating ammo…but the actual combat effectiveness of this is questionable, and most countries don’t trust it. AFAIK, only the British actually stock it much.
The runner up old stand by is HEAT, High-Explosive, Anti-Tank, also known as the shaped charge. This is what almost all rockets, recoilless rifles (bazookas), mines and missiles use, since it’s not reliant on velocity. This is a cone-shaped jacket of copper (pointy end away from the enemy) with lots of explosive packed behind it. Upon detonation, the explosive pushes the copper into a high-speed, narrow jet of molten metal, which burrows through the armor, sending armor and jet fragments into the crew compartment, turning the crew into jelly and so on (seeing a trend here? ) HEAT is losing favor of late, however, since the march of technology is rendering it relatively easy to defend against – you won’t do much to modern tanks with it (but personnel carriers are a different matter) – however, it’s still the large majority of anti-tank weaponry, due to its age-old simplicity, low cost, and ability to be put as an active warhead on pretty much anything.
The main AT round in use in WW2 was merely plain old AP, an Armor-Piercing slug (with a hardened nose) that just tried to bash its way into the enemy…basically just a giant bullet. Upon penetration, it would send fragments of the armor and round into the crew compartment, etc, etc. If it didn’t penetrate, it would ricochet, or just plain break up if the armor was harder than the round itself. This is what happened to most Shermans before the upgunning, their shells would simply shatter on the heavy tanks.
So, as you can see, no matter what weaponry is being used, they all have the same major goal: to poke a hole in the armor and send pointy bits flying around and into the crew. If you don’t penetrate the armor, you’re not really doing diddly squat to the tank, it’s usually an all or nothing affair. A side benefit of this is that, since tank kills are really usually just crew kills, the tank itself is often still fully operational (although you’d have to replace the electrical system and such if it burned) and can be reused after giving it a patchup and hosing the inside out. This was very common with Shermans, and they went to great lengths to hide this fact from the new crews.
There is, however, one major exception to this rule: that of the Really Big Shell. We’re talking large-bore artillery, like modern 155mm howitzers, or the 12-16 inch shells of WW2 naval bombardment, and aircraft-dropped bombs. A few kills by 120mm mortars were recorded in WW2, as well. Basically, these don’t penetrate the armor, they just totally obliterate it. They often hit the top armor, the weakest part of a tank, and crush the tank like a bug.
And that’s all I have to say about that
The missile itself isn’t causing that, it’s just the typical HEAT warhead poking a hole in it. That massive explosion is the “ammunition” inside the tank detonating. In fact, most people think it’s not actually tank ammunition, since the only description anyone’s gotten of what’s in the tank is that of being “similar to an appropriate combat load” – which probably means it was stuffed with the same amount of explosive as would be in the ammunition, but instead of being individually encased in 40 chunks, inside the autoloader, it was probably there in one big lump, and probably of more potent stuff. Kablooey! Not that it really matters, an ammo detonation of any scale is usually lethal to a crew (barring design tricks like the M1 has) and, in fact, the crew can be killed from an AT hit without any real visible sign of damage on the outside. It depends on where the hit is, how the tank is built, and what the weapon is.
