Since I can't know if readers understand the difference (you might, others may not), here's a simple example:
You have to lift 1000 kg of metal to the top of a massive 1000 meter tall skyscraper being built. It's a thousand 1 kg pieces. Let's say it takes 10 minutes to walk up to the top.
You can do this at least two ways:
One person moves each 1 kg piece up to the top, one at a time. That will require 1000 times 600 seconds, or 600,000 seconds. That's roughly seven days.
The other way is to do it all at once. We get a thousand people, each one of them grabs a 1 kg piece and, ten minutes later the entire load is at the top of the building.
Anyone can understand that these problems require vastly different scenarios.
Energy is about the total amount of work done. In this case it is the same for both scenarios.
Power, on the other hand, is a measure of how quickly this work was done. In this hypothetical, the fast delivery scenario required a thousand times more power than the slower version.
Electric cars require power. Energy is delivered over time. The problem with our grid and power generation is that it simply cannot handle this beyond a very small deployment. People driving electric cars today are enjoying the fact that they do not pose a significant threat to the existing infrastructure and generation capacity. As the numbers increase we will start having problems.
The idea of having a million or ten million cars simultaneously fast-charging today is likely an unthinkable reality.
Put a different way (just pulling hypothetical numbers out of my behind, these are not real numbers):
If we could charge electric cars over, say, 30 days, we might not need even one bit of additional power generation capacity or changes to the infrastructure. Sure, nobody would be able to drive more than once a month, but, hey, we don't need to build nuclear power plants.
However, that is not a practical reality. Tens of millions of cars will need to quick charge in 20, 40, 60 minutes. And another set of millions of cars will want to be charged over, say, 8 hours, overnight. This requires the ability to deliver power --large amounts of it-- in a way we are not prepared to do. This is both true in terms of power generation capacity as well as infrastructure, from the cables all the way to the power plants.
What the page you linked shows is an increase in energy consumption. It says nothing about power. Also, the increase you point out is only 60%, not a doubling from 1980 to 2020. Still, this is somewhat meaningless. I can consume 60% more energy if I keep the lights on 12.8 hours per day, instead of 8. I would not demand 60% more power out of the grid, I would just use the available power for a longer period of time. That's why speaking in terms of energy isn't relevant to the electric car problem. The problem is power, not energy. People need their cars charged now, right now. That requires power we cannot deliver.