For a given length of barrel, the heavier profile barrel will be stiffer.
Barrel harmonics are improved.
Also, the heavy barrel's increased mass will allow it to absorb more heat before overheating. Good for full-auto, and rapid shooting for longer time periods.

Yes, the greater mass of the heavy barrel will take longer to cool, but it also has greater surface area from the larger diameter. That increased surface area will allow a faster cooling rate than a skinny barrel, so the actual time for cooling may not be as vastly different as some surmise.

Most people who understand what different barrels can do, will know what barrel they want.

The rest of the market is influenced by "fads" which come and go. 10 years ago, everyone wanted heavy barrels and you couldn't give away a gov't profile barrel. 2 years ago, most people wanted the lightweight barrels, and heavy barrels didn't sell nearly as well, and some people went so far as to say that they were only made for cost-saving purposes to save the manufacturer money on machine time.
Now, I'm seeing a trend toward "midweight". Probably because after the lightweight trend, people experienced the shortcomings of the lightweight barrels on their last build, so they are "inching" back toward the heavy barrel.