To answer your questions, I will provide 3 different solutions to the same problem.

Case 1: just normal Python

import time

def sleep():
    print(f'Time: {time.time() - start:.2f}')
    time.sleep(1)

def sum(name, numbers):
    total = 0
    for number in numbers:
        print(f'Task {name}: Computing {total}+{number}')
        sleep()
        total += number
    print(f'Task {name}: Sum = {total}\n')

start = time.time()
tasks = [
    sum("A", [1, 2]),
    sum("B", [1, 2, 3]),
]
end = time.time()
print(f'Time: {end-start:.2f} sec')

output:

Task A: Computing 0+1
Time: 0.00
Task A: Computing 1+2
Time: 1.00
Task A: Sum = 3

Task B: Computing 0+1
Time: 2.01
Task B: Computing 1+2
Time: 3.01
Task B: Computing 3+3
Time: 4.01
Task B: Sum = 6

Time: 5.02 sec

Case 2: async/await done wrong

import asyncio
import time

async def sleep():
    print(f'Time: {time.time() - start:.2f}')
    time.sleep(1)

async def sum(name, numbers):
    total = 0
    for number in numbers:
        print(f'Task {name}: Computing {total}+{number}')
        await sleep()
        total += number
    print(f'Task {name}: Sum = {total}\n')

start = time.time()

loop = asyncio.get_event_loop()
tasks = [
    loop.create_task(sum("A", [1, 2])),
    loop.create_task(sum("B", [1, 2, 3])),
]
loop.run_until_complete(asyncio.wait(tasks))
loop.close()

end = time.time()
print(f'Time: {end-start:.2f} sec')

output:

Task A: Computing 0+1
Time: 0.00
Task A: Computing 1+2
Time: 1.00
Task A: Sum = 3

Task B: Computing 0+1
Time: 2.01
Task B: Computing 1+2
Time: 3.01
Task B: Computing 3+3
Time: 4.01
Task B: Sum = 6

Time: 5.01 sec

Case 3: async/await done right

Same as case 2 except the sleep function:

async def sleep():
    print(f'Time: {time.time() - start:.2f}')
    await asyncio.sleep(1)

output:

Task A: Computing 0+1
Time: 0.00
Task B: Computing 0+1
Time: 0.00
Task A: Computing 1+2
Time: 1.00
Task B: Computing 1+2
Time: 1.00
Task A: Sum = 3

Task B: Computing 3+3
Time: 2.00
Task B: Sum = 6

Time: 3.01 sec

Case 1 and case 2 give the same 5 seconds, whereas case 3 just 3 seconds. So the async/await done right is faster.

The reason for the difference is within the implementation of sleep function.

# case 1
def sleep():
    ...
    time.sleep(1)

# case 2
async def sleep():
    ...
    time.sleep(1)

# case 3
async def sleep():
    ...
    await asyncio.sleep(1)

In case 1 and case 2, they are the "same": they "sleep" without allowing others to use the resources. Whereas in case 3, it allows access to the resources when it is asleep.

In case 2, we added async to the normal function. However the event loop will run it without interruption. Why? Because we didn't say where the loop is allowed to interrupt your function to run another task.

In case 3, we told the event loop exactly where to interrupt the function to run another task. Where exactly? Right here!

await asyncio.sleep(1)

More on this read here

Update 02/May/2020

Consider reading

• A Hitchhikers Guide to Asynchronous Programming
• Asyncio Futures and Coroutines

is it possible to give a simple example showing how async / await works, by using only these two keywords + asyncio.get_event_loop() + run_until_complete + other Python code but no other asyncio functions?

This way it's possible to write code that works:

import asyncio


async def main():
    print('done!')


if __name__ ==  '__main__':
    loop = asyncio.get_event_loop()
    loop.run_until_complete(main())

But this way it's impossible to demonstrate why you need asyncio.

By the way, why do you need asyncio, not just plain code? Answer is - asyncio allows you to get performance benefit when you parallelize I/O blocking operations (like reading/writing to network). And to write useful example you need to use async implementation of those operations.

Please read this answer for more detailed explanation.

Upd:

ok, here's example that uses asyncio.sleep to imitate I/O blocking operation and asyncio.gather that shows how you can run multiple blocking operations concurrently:

import asyncio


async def io_related(name):
    print(f'{name} started')
    await asyncio.sleep(1)
    print(f'{name} finished')


async def main():
    await asyncio.gather(
        io_related('first'),
        io_related('second'),
    )  # 1s + 1s = over 1s


if __name__ ==  '__main__':
    loop = asyncio.get_event_loop()
    loop.run_until_complete(main())

Output:

first started
second started
first finished
second finished
[Finished in 1.2s]

Note how both io_related started then, after only one second, both done.

Python 3.7+ now has a simpler API (in my opinion) with a simpler wording (easier to remember than "ensure_future"): you can use create_task which returns a Task object (that can be useful later to cancel the task if needed).

Basic example 1

import asyncio

async def hello(i):
    print(f"hello {i} started")
    await asyncio.sleep(4)
    print(f"hello {i} done")

async def main():
    task1 = asyncio.create_task(hello(1))  # returns immediately, the task is created
    await asyncio.sleep(3)
    task2 = asyncio.create_task(hello(2))
    await task1
    await task2

asyncio.run(main())  # main loop

Result:

hello 1 started
hello 2 started
hello 1 done
hello 2 done

Basic example 2

If you need to get the return value of these async functions, then gather is useful. The following example is inspired from the documentation, but unfortunately the doc doesn't show what gather is really useful for: getting the return values!

import asyncio

async def factorial(n):
    f = 1
    for i in range(2, n + 1):
        print(f"Computing factorial({n}), currently i={i}...")
        await asyncio.sleep(1)
        f *= i
    return f

async def main():
    L = await asyncio.gather(factorial(2), factorial(3), factorial(4))
    print(L)  # [2, 6, 24]

asyncio.run(main())

Expected output:

Computing factorial(2), currently i=2...
Computing factorial(3), currently i=2...
Computing factorial(4), currently i=2...
Computing factorial(3), currently i=3...
Computing factorial(4), currently i=3...
Computing factorial(4), currently i=4...
[2, 6, 24]

PS: even if you use asyncio, and not trio, the tutorial of the latter was helpful for me to grok Python asynchronous programming.