You'll see a lot more difference going from a Marlin or Ruger to something like a CZ, than you would going from a standard profile barrel to a heavy barrel within the same model.
Having said that, two of the advantages of a heavy barrel are heat dissipation and inertial dampening.
A heavier barrel has more material to heat up, so it takes a lot longer for you to see point of aim vs. point of impact drift, which mostly matters if you're firing a lot. This can actually matter for high volume varminting. I've gotten barrels screaming hot working over a gopher town (shooting 100+ gophers in an hour), but this is the exception, not the rule. Generally, even on a pretty active day, there will be enough time between shots that a standard profile barrel won't heat up to any significant degree.
Inertial dampening with heavy barrels might not be what you think it is. With a longer, heavier barrel, it takes more effort to move it around (inertia), so it tends to dampen the minor movements created by the shooter. I use an old (late 40's manufacture) Remington Matchmaster from time to time, and in the field, the thing just seems to be a laser, even if it is a bit of a pain to lug around - it's a big, heavy, rifle for a .22. But put it in a vice on a bench, and pop off groups next to a thin barrel CZ 452, and the CZ actually shoots a bit better. The difference is that the CZ, while being lighter and handier, because of it's light weight and balance is "twitchier" - basically it's more prone to shooter error (those minor motions from your heart rate and breathing).