Not to complicate matters but there are 2 different definitions of fast action – traditional and modern.
Modern fast action rods tend to be far too stiff with tips that are too stiff to protect tippets and these rods really aren’t useful trout fishing rods at anything under 60 feet. These rods are not designed for typical trout fishing situations (under 40 feet) but rather launching the entire line or for salt or heavy duty fresh water fishing. These rods generally need to be over lined to be able to fish at distances under 40 feet and a well made fast action should not need to be over lined to fish at closer distances. Examples of this kind of cannon rod are the Sage TCX and ONE, and the Temple Fork TiCr, TiCrx and Axiom (I’m sure there are others but these come to mind). These are not trout fishing rods. Another thing is most cheaper rods tend to follow this design because it is cheaper to manufacture this type of rod than a traditional fast action rod because making a traditional fast action rod requires smaller mandrels and finesse in making the tip section – the most critical part of a fast action rod – and that adds money.
Traditional graphite fast action rods have a soft tip section (maybe 24” on a 9’ rod) and they get progressively stiffer as you approach the butt section. The soft tip protects tippets and allows you to make delicate presentations at close ranges by casting off the tip. It is a fine balance between a soft tip and stiff tip because too soft, while good at close ranges, affects accuracy at distance. This was precisely the problem with the original Winston BiiX – the tip section on certain models was far too soft and you simply couldn’t cast accurately at distances over 40 feet or so. Good examples of a traditional graphite fast action rod are the Winson BiiiX, Sage Z-Axis, Hardy Zenith and Loomis NRX. Even between these 4 rods there are differences with the Winston excelling best at close ranges but maxing out at about 65 feet while the NRX has the least feel of the 4 at close ranges (can do the job) but has the most feel of the 4 at distances greater than 40 feet and also launches the most line. The Z-Axis and Zenith are probably the best rods of the 4 for “across the range performance,” from close range (but not quite as good as the Winston) to long distances (but not quite as good as the NRX). I would also throw in the Temple Fork Professional Series as being a pretty good example of this type of rod.
Medium and medium fast action rods are basically a progressive action rod like a fast action rod except the softer tip extends further down the rod (maybe 1/3 to 1/2) before becoming stiffer. Good examples are the Winston WT and BiiT, Sage ZXL, Loomis Whisper Creek, Scott G2 and Temple Fork Finesse.
Slow action rods actually have stiffer tips and butt sections and bend more in the middle section, sometimes calls parabolic action. You won’t find many graphite rods of this design and most of these are older or retro fiberglass or boo rods.
As for fishing applications, FishIdiot’s first paragraph pretty much gives a good summary. If I know I am going to be fishing a hatch on your typical PA stream (fishing under 40 feet), I prefer a medium action 9’, 5wt. If I will be fishing predominately subsurface, I prefer a fast action 9’, 6wt.
Overall most people prefer fast action rods simply because they are more forgiving and compensate for less than perfect casting technique and timing. Medium and slow action rods require more precise timing and better technique and if you don’t have it, your will struggle to cast more than 25-30 feet of line. But in the hands of an expert caster, that person will be able to cast either rod action equally well.
My suggestion if thinking of buying a new rod is what is the purpose of the new rod? How will it be fished? Far too many people walk into a store, grab a rod, and start launching 70 feet of line and they’ve never been able to do that before so they buy the rod only to find out they can’t cast it under 40 feet without over lining it. Cast and test rods under the circumstances you will be fishing.