I apologize for the long post but when discussing casts we need to be sure we use the same terms. Otherwise, there is confusion. I get really frustrated when I see what are the incorrect terms used to name casts. I believe casting terms are loosely used and often times incorrectly.
So here is my version of what is correct and what is not.
Mato,
I don’t think that is a pile cast. Another case of the internet spreading the wrong information. I think that is what one would call the bounce cast, where the forward cast is overpowered or shocked with a haul at the stop. This overpowered cast “bounces” the fly, leader, and line back and it falls in “S” curves. It is like the Wriggle or Wiggle cast. See this description or the Wriggle and the bounce from Sexyloops:
http://www.sexyloops.com/flycasting/twriggle.shtml
Here is a video that shows it.
http://video.answers.com/advanced-fly-casting-techniques-the-slack-line-cast-276850312
The puddle cast is also called the pile cast which confuses it with what I consider the pile cast. First of all, the puddle “cast” is not really a cast. It is a downward mend that occurs after the cast. So the more proper term is the puddle mend and pile mend to distinguish it from the tuck cast and pile cast which are real casts.
As we know, the end of a cast is defined by the rod stop. What happen afterwards by moving the rod or the rod tip is a mend and not the cast. A reach mend is a reach to the side after the rod stop. A puddle mend is mend down after the stop.
It is a low back cast combined with a high forward underpowered cast with an open loop. After the stop, you immediately lower the rod tip (a downward mend), and the underpowered cast collapses and the leader and line puddles on the water over the target.
The descriptions below from the internet do not mention the underpowered nature of the cast, but the pile will contain more slack with an underpowered open loop. You can also shoot line to kill the cast at the stop and drop. It is a cast that barely extends and falls back on itself. The sites call it the puddle or pile cast but as I’ll show later it should be called just the puddle with the term “pile” cast reserved for a version of the tuck with a dry fly. Puddle cast:
http://www.sexyloops.com/flycasting/tpile.shtml
http://www.virtualflycasting.com/cast%207c.htm
http://www.flyfishingyellowstonenationalpark.com/ai455.html
I described the tuck and pile casts in a previous post. These are true casts that are created before the stop. You can combine them with mends after the stop as I noted in my original description of an up and forward mend after the stop. The text and photos below are from Jason Borger’s book The Nature of Fly Casting.
Text below is from pp 190 and verifies what I stated before about the two casts.

Here is the illustration of the two casts. The Tuck is the more solid line than flips completely over. The Pile shows the trailing distal leader and tippet with the trailing air resistant dry fly.

When one can perform both casts, there is an advantage to the pile cast over the puddle mend. Because the puddle mend is a high trajectory and depends on gravity to puddle the line/leader it is less accurate than the pile. Any wind or gust can make the puddle useless.
Secondly, I use this cast when casting over a section of fast water to a slot of feeding fish behind or ahead of a rock. On the puddle, the line closest to you falls to the water first and this means the line is being dragged downstream before the fly lands. You get a shorter and less accurate drift with the puddle.
On the pile cast, the leader lands first and before the line so you get the fly on the water before the line hits the water and pulls on the leader.
So use the pile cast when you can.
Why would you not always use the pile? The problem is distance. I can make a puddle at a distance that is too far for me to do a pile with my level of skill.
Here is a photo of place where the pile or puddle are the only two casts that will work. The fish are feeding in seam “A” and you need to cast from the bank at “B” across that fast water.

Here is another place here the fish feed in the slot B or on the other side of the A . The fish also feed in front of the rock and here the pile provides better accuracy.
