There are basically two ways that electricity kills you.
The first is that it can screw up your heartbeat. As you noted, this takes very little current. Most safety standards are built around under 5mA being the “safe” current level. Once you get up around 50 mA the risk becomes pretty significant. The thing is, even at its best, this is kinda hit and miss. The heart is more sensitive at certain times during its rhythm than others to disruption, so when the shock hits relative to the heart’s rhythm matters. This type of shock tends to throw your heart into fibrillation, and your heart has kind of a funny design in that this fibrillation state is stable, meaning that the heart will happily stay in fibrillation until something else gets it out of this state. Since your heart is just kinda shaking and isn’t really pumping, you pass out fairly quickly and then die from lack of blood flow.
It’s a bit counter-intuitive, but as you increase the current, at some point the risk of death actually drops significantly. What happens is that instead of going into fibrillation, the heart muscles just clamp. At that point the heart isn’t pumping blood, so if the source of electricity isn’t removed you’ll still die. But, unlike in fibrillation, if you do remove the current the heart that was clamped will usually go back into a normal rhythm.
Above that level of current though, you start to get into the second way that electricity kills you. It quite literally cooks you to death. Electricity through pretty much anything other than a superconductor creates heat. Electricity flowing through your body will create heat and will cook your brain and internal organs. This is how an electric chair kills you, and unlike the fibrillation method, this one is much more reliable and isn’t so hit and miss. People don’t tend to survive the electric chair so it’s difficult to confirm this, but the theory is that the first jolt fries your brain to the point where you are no longer conscious. You then aren’t aware of your body being cooked to death by the electricity. At some point your heart gets so badly damaged from all the heat that it stops beating, and you are declared dead. Theoretically, your brain was toasted long before that.
By using a dry sponge, there wasn’t a good connection for the electricity, so the amount of current flowing through the body was greatly reduced. Now, it’s not so certain that brain was toasted by the first jolt, and he may have been conscious for at least part of the execution. If he was conscious, it would have been extremely painful, and possibly not fatal if the electricity had been removed at some point. As it was, he died a slow, painful death where his body was slowly cooked to death. It would have been nice if his heart went into fibrillation as that would have rendered him unconscious in about ten seconds or so, but fibrillation isn’t guaranteed even under those circumstances. It’s also not guaranteed that his heart muscles would have clamped from the electricity, which would have similarly rendered him unconscious due to lack of blood flow.
While he likely wouldn’t have stayed conscious for the entire thing, it would have been absolutely horrific for the spectators, as they would have smelled burning flesh from the bad connections, and the entire execution would have taken significantly longer than usual. Even though he wasn’t conscious, the body would likely still have been twitching and thrashing, which again is more horrifying for the spectators than the victim since the victim probably wouldn’t be conscious for most of that.