What a wonderfully macabre question! This calls for cited sources!
First off, I'd like to point out that you are correct to notice the similarities between a grenade and a rifle round. Both have the same gross pattern: they set off some explosives near some metal that is going to be accelerated by the explosive. So if we start from that very simplified model, we can cover both the similarities and the differences between the two.
In both cases we start with a detonation. This is a combustion reaction which is fast enough to propagate a shock wave in front of it. The first thing we can do is contrast this with deflagration, where the combustion proceeds slower than the speed of sound. In both grenades and modern firearms (which use a nitrocellulose based explosive typically called "smokeless powder"), the reaction is a detonation. However, in black powder based firearms from previous eras or lower quality improvised grenades, the reaction is a deflagration. I'll assume that we are interested in the modern military grade equipment, given the wording of your question, so if you're interested in the deflagration case, simply skip the sections dealing with the direct consequences of detonation.
(Thanks to all the commenters posting on this, I've had to do more research and issue a correction. Explosives detonate, which is defined to be what happens when a combustion reaction propagates faster than the speed of sound. The propellants in ammunition are designed to deflagrate, which is burning with a flame front that moves slower than the speed of sound. My confusion was caused by the presence of supersonic bullet, which would clearly need to be propelled by something going faster than the speed of sound. The piece I missed was answered here: the high temperature of the expanding gasses increases the speed of sound in that medium. Also, the speed of sound that matters is apparently the speed of sound through the material (the powder itself), not the air surrounding the powder. That speed of sound is much faster. Firearms designers use this deflagrating "low explosive" intentionally, for many reasons including the fact that this means their propellant cannot generate the shock waves of a high explosive, and thus does not need to be regulated in the way we regulate high explosives. I can go to the store and buy smokeless powder without any questions. If I went to the store and asked to buy C4, questions would start coming my way. Thanks to everyone for pointing out my mistake!)
When you detonate an explosive, you create a shock wave. Shock waves are interesting little beasts.
If you want to skip the next few paragraphs on the physics of a shock wave, you can. However, I find its very helpful to understand what a shock wave actually is. That helps in understanding why the line between deflagration and detonation is such a big deal.
A shock wave is required because the simplistic laws of physics that we're used to break down. Usually we say that information about an object propagates ahead of it at the speed of sound. What this really means is that there are gas atoms which hit the object and are bounced in the opposite direction, traveling faster than the object itself. These eventually collide with other gas atoms, sending them scattering ahead of the object and so on and so forth. In normal every day circumstances, this process involves so many collisions that we can model it statistically. The result is that we can talk about "pressure" meaningfully, and we can talk about a "pressure wave" which propagates ahead of the object. All objects moving through the air have a pressure wave in front of them, though it is not always obvious. A fast moving modern car doesn't have very many insects striking its windshield because the pressure wave in front of it pushes the insects upwards over the windshield. An older car with worse aerodynamics may not generate a sufficient pressure wave to force the insect over the top of the car, and the result is... well.. messy.
When we get to events that move at the speed of sound or faster. As this happens, our nice clean statistical model of the gas breaks down. In the nice clean model we're used to, we say that every small region has a "pressure," and it pushes outwards in all directions equally. This works because, over the distances between collisions (the "mean free path length," on the order of 68nm), the velocity of particles on both sides of the object are close enough to use easy statistical distributions. However, as particle speeds approach the speed of sound, this changes. The differences in velocity get more and more pronounced until we can't sweep the differences under the rug with a simple differential equation. We have to account for more of the actual particle physics in this regime.
If you're skipping the physics lesson, this is a good time to rejoin the answer
So why is a shock wave such a big deal? Well when we account for the particle physics of objects at these speeds, we have to allow for what are basically discontinuities in pressure (modern measurements suggest a shock wave is about 200nm thick). The pressure can rise almost instantly. This matters because many objects, including human bodies, are very sensitive to sudden changes in pressure. Under normal conditions, such rapid pressure changes would call for an enormous amount of energy. However, if you can create a shock wave, you can create a rather large pressure differential with a much smaller explosive.
So what happens when a shock wave hits the body? For the most part, it can pass through freely, but if we find regions where the speed of sound changes dramatically, these sharp pressure changes can do damage.
The primary effect of such shockwaves is on the lungs. This effect is dominated by the impulse of the explosion (generally proportional to the integral of overpressure over the duration of the shock). In the case of the lungs, this impulse imparts a velocity to the cells lining the alveoli. Because they are thin, designed to stretch as we breathe, and right along an impedance boundary between flesh and air, they are very susceptible to this. The result of this damage is the rupture of the capillaries in the lungs, called pulmonary contusion. This is what medical doctors would call "Bad News." The damage can cause the alveoli to collapse, no longer participating in breathing. It can also cause pulmonary edema, filling the lungs with fluid and causing suffocation.
Other gas filled organs can be affected similarly, but in the literature, lung damage due to shocks is the primary issue.
Ear drum damage also occurs in response to such shocwaves. Being thicker than the walls of the alveoli, they respond to slower effects. The damage to the ear drum appears to be more associated with the actual overpressure than the impulse of the shockwave. A perforated eardrum is typically not fatal, but it can cause sudden disorientation and vertigo. Given that one is in an environment where there are grenades and/or flying bullets, this disorientation can lead towards a fatal incident shortly thereafter.
So this shows a major difference between grenades and bullets. In a bullet, the shock wave occurs quite far away and has rather low impulse by the time it arrives. The shockwave plays a very minor part. There is an argument that a supersonic bullet hitting flesh can generate its own shockwaves which can disrupt neural activity, known as hydrostatic shock, but it is a disputed theory.
Thus we see a major difference between the bullet proof vest and the EOD suit. You noticed that it covers the whole body, but it also covers it in a different way. The layers of an EOD suit are also designed to redirect and damp the shockwave. They do this using layers of varying acoustic impedance. EOD suits are also designed to protect in other ways, such as cushioning the spine so that an EOD expert thrown back by an explosion is unlikely to suffer catastrophic spinal injury.
Now some grenades stop here. Concussion grenades like the MK3 do their damage with these effects. A fragmentation grenade like the M67 adds a layer of metallic shrapnel. This shrapnel operates like a bullet. In fact, it is reasonable to model the effects of shrapnel exactly like we model the effects of bullets.
Bullets are really straightforward. Shove a metal slug through someone's body, and you force the bonds that hold their body together to give way. If any of those bonds were critical, the opponent is incapacitated.
Bullets come in supersonic and subsonic varieties. The fundamental difference between them would be that a supersonic bullet could cause a shockwave to propagate through the body. However, given that hydrostatic shock is a disputed theory, we can reasonably ignore that difference. Instead, we can just look at all bullets as the same sort of thing. Their damage is based on shape, energy, and momentum. Naturally, supersonic bullets can have substantially more energy, but other than that they aren't special.
A bullet entering a wound basically generates (subsonic) waves, pushing the flesh out of the way of the bullet just like the air was pushed around our windshield in the car example at the beginning of this answer. This pushing effect can tear tissue, and that's the primary cause of damage from a bullet (or grenade fragment).
If arteries, veins, or capillaries burst, blood loss will occur and may cause death. Damage to nerves can cause paralysis of the innervated region, and obviously damage to the brain can cause death. A bullet may break a bone, in which case those muscles can no longer effectively use that bone to create motion. It may also tear tings like tendons, which also prevent motion.
If a bullet or fragment strikes any area, it may cause infection. This is a major factor in abdominal wounds. Our intestines are quite full of bacteria kept safely within the body of the intestines. If the intestines are torn, they will spill this material out, creating a substantial risk of infection.
The kevlar and/or ceramics found in both bullet proof vests and EOD suits is focused on dealing with these objects. Both materials are very good at arresting physical objects before they enter the body. Bullet proof vests have a smaller coverage area because of tradeoffs. Those who wear bullet proof vests must move quickly and care about minimizing burden. Thus the vests only cover the regions where the lethality of a bullet wound warrants the burden of protecting it.
In the case of the EOD suit, mobility is less of a concern. The EOD technician is already where they need to be (which would be the place everybody else doesn't want to be). They do care about mobility, don't get me wrong, but the tradeoffs for someone intentionally going to the wrong sort of place are different. It's worth it to them to have full body coverage.
Which leaves me with a gem of wisdom I got from the comic Schlock Mercenary, by Howard Taylor. His The Seventy Maxims of Maximally Effective Mercenaries includes two which I am yet to find a veteran or active duty member who doesn't agree with, or at least have to give a nod at the wisdom of it all:
- A Sergeant in motion outranks a Lieutenant who doesn't know what's going on.
- An ordnance technician at a dead run outranks everybody.