Let me give you an example. I have a Big Green Egg controlled by a Stoker. This morning before I left for work I put on a Pork Butt and I am monitoring the cook via the internet from my office (Stokerlog). I couldn't do that with a stick burner. However, I really like tending the fire in a stickburner and would prefer to do that if it were a Saturday afternoon and I had nothing better to do. I also just bought a drum to make a UDS so I could have something to drag around on Boy Scout campouts without fear of damaging my more expensive cookers.
But it appears that yes, different cookers work differently on different meats. Recently some people have found that they like chicken better from a vertical smoker where they can hang the chicken halves, but prefer the taste of brisket in an offset smoker. The right tool for the right job.
The best advice I can give you is master one smoker before you move on to the next.
Large BGE - WSM - Small Charbroil Offset - Weber Performer