You asked for input from stick burner users so here goes . . .I cooked on a stick burner for many years (owned several brands) and frequently had problems with oversmoking and maintaining temperatures reliably. Thus, I was one of the first to try pellet cookers when they appeared. I almost gave up because my first, a used Traeger, was a basket case but I didn't know it. Kept at it. Now I'm hooked on pellets but occasionally still use my charcoalers, too (because I like charcoal flavor).
If you like heavy smoke flavor, you're never going to be happy with a pellet cooker unless you augment it with a smoke generator of some type. Even then, if you have to have really heavy smoke you won't be satisfied. See my articles here on that subject.
The kind of barbeque we like is a matter of personal taste and one should have the equipment that produces it. There's nothing wrong with heavy smoke if that is what you like. As a judge I dread it because I know I'll be tasting it the rest of the day and there's no getting rid of it -- the water and crackers won't cleanse it from the palate. Oversmoking is a deal-killer in competition where the goal is to produce a unique, positive blend of flavors from all the components). If I get a heavily smoked sample (doesn't happen too often) early I'll have trouble clearing the smoke influence from my tastebuds for the next entries. Same goes with super hot (extremely rare in competition) and over-sauced (fairly common).
A smoke ring on meat is a chemical reaction, not an indication of flavor or flavor intensity. If you like smoke rings, get a box of Morton's Tender Quick and rub a little on the meat or use it as a component in your rub recipe (careful, it is quite salty). It'll produce a real halo.
As to tasting the difference between or among wood types on a stick burner vs. a pellet cooker that's probably a function of smoke intensity. I have found it depends, to a degree, on the meat. Chicken, for instance, will offer up a different flavor profile when cooked with sharp/strong pellet types (hickory or mesquite) vs. lighter, sweeter woods like apple or cherry. Meats with a more distinct and stronger natural flavor will not alter their flavor profile much due to smoke type. Heavy smoke application (of any wood type) should "reinforce" that flavor enough that you can tell the difference, maybe even by just the smell of the meat alone.
Cook what you like! There are lots of different outdoor cooking fuels and methods and somewhere among them is the perfect method for everybody
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