*snip*
See that gate on the left? The part that has the "-" at the top (where god intended it) indicating that if you bump the lever upward your car will downshift? That's a very important part of driving.
It has come to my attention that very few people know how to use this feature of their cars and, as a result, are not driving their cars properly.
Outside of using it for engine braking, I pretty much never touch the manual mode in my car ('08 Mazda6) anymore. There's simply no point.
Scenario
You're driving on the freeway and you're stuck behind a slow moving vehicle going 10 mph slower than the other lanes. You want to get out from around this vehicle. How do you do that properly in an automatic?
Signal, see gap, nail throttle, sorted. Pretty much the same moment my foot hits the floor the car has executed the downshift to 2nd or 3rd (depending on speed). If I'm not being my usual Jerkass self then slightly less throttle will result in only dropping to 3rd or 4th (again, speed-dependent) but with the same result.
Observe a break in traffic approaching you in the overtake lane. Before it arrives, move the stick from automatic mode into the manual mode and select a downshift.
Why? It's much slower to do that than just tap the gas.
This will cause your car to shift into a lower gear and give you power for overtaking.
So will hitting the throttle as you normally would in an automatic.
When the gap arrives, you are already in the proper gear and can simply accelerate into the merge. After you have completed merging, slide the lever back into automatic mode.
Already being in the proper gear means pretty much nothing when the vast majority of automatics will drop from top gear to as low as 2nd pretty much instantaneously.
How does this differ from what you're used to. What you're used to is using the throttle (a total different control) for modulating the transmission.
Yes, and?
As a result, you have to mash the throttle to the appropriate depth to try to force the transmission to downshift. When you do that, you not only have to wait for the transmission to shift (losing valuable seconds as the gap passes you by) but you also commit yourself to accelerating toward the vehicle in front of you until you merge.
Valuable seconds? The
absolute worst-case scenario is an older automatic (mine included) designed in a way that means there is a slight (usually around one second) delay between the request for 1st gear and the transmission actually going through with it. This downshift will only happen at rather low speeds (generally under 20mph) and there is absolutely no avoiding how slowly it will happen. The rest of them, well... You're in the gear your throttle position asks for pretty much the exact moment you ask for it.
By selecting the gear ahead of time, you have significantly better control and don't have unintended consequences.
I certainly don't see it. I don't see how I'm "committed" to accelerating towards the vehicle in front of me (as I can obviously just let off), I don't see the time argument whatsoever (and in fact it could be argued you're "distracted" longer using the manual mode than not using it quite easily), I don't see any of it.
Scenario
You have to slow down for a turn, but you know you'll have to get on the gas hard after the turn. How do you do this properly in an automatic?
Choice 1: Hit throttle slightly before you want full throttle.
Choice 2: Manually select lowest possible gear for planned exit speed, proceed as you would in a manual.
The answer is that while you're braking you move the lever into manual mode and select a downshift (or two). That way when you're ready to accelerate the power is available for you. If you simply wait until you're ready to accelerate you will have a delay while the transmission tries to find the appropriate gear.
Delay of approximately a tenth of a second. Valuable on a race track but on the street it really isn't a huge deal.
The gear selection will be rougher
Considering automatics rev-matching manual downshifts is a
very recent thing, it's really about the same.
you will back up traffic behind you (briefly)
lolwut?
lolwut?
and you may not get the gear you want if your right foot didn't find the right place in the throttle to convince the transmission to select the gear you wanted.
You said accelerate hard. Accelerate hard means an auto will find the lowest possible gear and use it. If you wanted anything else, you didn't want to accelerate "hard", you wanted brisk, and your throttle usage would reflect it.
Scenario
Your car is underpowered and you're going up a long hill. The transmission constantly wants to select a gear that is either too tall, or too short for the hill. As a result, you pop back and forth between 2nd and 3rd gear all the way up the hill. How do you handle this properly in an automatic?
The answer is to slide the lever into manual mode and hold gear 2 (the lower gear that the transmission was selecting). This eliminates needles shifts, inconsistent speed, is less irritating for drivers behind you, is smoother for your passengers, and gives you the power to accelerate more if you need to. Remember that this hill requires this gear and do it in advance next time you're on this hill. After you get up the hill, slide the lever back into automatic mode.
I've only had this happen with old-school transmissions with no separate manual mode, and have only had them hunt between overdrive and 3rd. Hit OD off or pop 3rd on the shifter depending on which setup it is, carry on. If you're going up a hill and the trans is continually hunting, you're 90% likely doing it wrong with the throttle.
The manual mode of your automatic is your friend. It is a tool. it is not something you need to use for every shift (it would be annoying and pointless to do so), but it is also not something to be forgotten. Develop the technique, and use it in your everyday driving. It should become second nature for you to realize that a given scenario is one of the few times automatic mode is not appropriate, and move naturally into manual mode.
As I said earlier, the only time mine sees any use is when I want to select a lower gear for engine braking. Oh, there's also the (rare) times when I want to use it to hold the shift off until just before the limiter instead of 500rpm before it.
Also, you do not need a manual mode like the one pictured above to make this happen. I have used this technique in a car much like this one:
The button on the side of the gear lever does not have to be pressed for you to slide it forward into D3 for a commanded downshift.
There is a reason for this, and the reason is that you're expected to need to do so. When you're done passing, slide it back into D4. It's a simple forward or back motion, no button pressing or unlocking of the lever.
As mentioned, the button DOES need to be pressed to go to D3 from D4 but not from D3 to D4. Same goes for each selection lower than that. Also, you sorta slide it forward from D3 to D4, which is an upshift. It also doesn't prevent a further downshift (to, say, 2nd) if you nail the throttle... Unless it's a Ford, in which case manual 2nd is in fact manual 2nd at all speeds, there is no "D3" (and instead an "overdrive off" button, which makes drive act as D3 in something else would).
. Last year, I made the horrifying transition from having a MTX daily driver to an ATX. The vehicle I wanted--and it was perfect for everything that I wanted it to do--was not offered with a manual. My CX-5 has the 6-speed autobox with manu-matic mode. Never having had an automatic before, I did find that I had to "learn" how to drive an automatic, in order to be as smooth as before. The manual mode is perfect for many situations, and once you know where the shortfalls are in fully-auto mode, you can avoid those annoying "what gear do I want?" moments that seem to baffle automatic transmissions. 
