There's a reason why most weightlifters in the Olympic Games don't look as polished as the bodybuilders you see in magazines and bodybuilding competitions. In order to have an aesthetically balanced and "complete" physique, bodybuilders not only have to build up the main lifting muscles, but also others such as the rear deltoids that are not normally used for heavy lifting. Therefore an impressive-looking bodybuilder can actually be weaker, in practical terms, than someone who has the same amount of muscle mass but concentrated in the places that really matter. A professional weightlifter's goal is lifting a lot of weight, so they concentrate on the muscles which have utilitarian purposes.
Bodybuilders don't merely have to build their muscles; to prepare for a competition they have to lose fat and water weight so that their muscles will be defined and visible. However, strength is determined by the muscles themselves, not whether there is fat as well; losing it is mostly a matter of showmanship for bodybuilders.
In order to perform a heavy compound lift such as a squat, deadlift, or clean and jerk, the core musculature of the lower back and midsection have to be well developed. This increases the thickness of the midsection with muscle even before you add any fat: note for example how strongman Mariusz Pudzianowski, who competed at low enough body fat percentages to have a fairly crisp six pack, had some massive obliques which gave a blocky look to his midsection. "Spilling over through the obliques" so that they protrude over the posing trunks and make the waist look wider is a known cosmetic issue for some bodybuilders, and those who are worried about it may deliberately avoid or downplay certain lifts in order to ensure they keep their waists streamlined.
In order to be able to lift with the intensity seen by strongmen and powerlifters, one must also consume absolutely massive quantities of food. Bodybuilders also do this for the sake of gaining mass, and if you look at them in the offseason, many of them appear quite chubby. It's only through starving and even dehydrating themselves prior to contests that they achieve their cut look. A bodybuilder is actually entering a contest at his weakest state, to the extent that sometimes a bodybuilder will start to feel lightheaded or even need to be carried off the stage for treatment, as famously happened to Paul Dillett at the 1994 Arnold Classic. Champion bodybuilders enter contests at a body fat percentage of around 5%, which is physically impossible to maintain for anything more than a day or two. A weightlifter, not being judged by his appearance, but rather his performance, will not take these strength-sapping steps.
Bodybuilding has increasingly fallen victim to this trope since Dorian Yates ushered in the Mass Monster era in 1993, as the the increased emphasis on size has come at the expense of the tight midsections and pleasing proportions which were common during the Golden Era. Contestants may still be able to shed their off-season fat and stand on stage shredded at up to 300 pounds, but when they get that large the muscles of the abdomen also get thicker and made the midsection look blocky, as mentioned before. Worse has been the "bubble gut" phenomonon which began to appear in the nineties and became widespread in the 2000s, where a bodybuilder's stomach will look bloated or even pregnant on stage despite them being at extremely low body fat. Various explanations have been put forward, some more likely than others. The use of human growth hormone has been blamed, with some speculating that it causes the guts and organs to literally grow in size; this cannot be correct, as internal organ growth is permanent, and bodybuilders such as Ben Pakulski and Roelly Winklaar have proven it is possible to cure or reverse the bubble gut (although exactly how they did it is a matter of speculation). One of the more plausible causes is bodybuilders overeating and relying on insulin injections to absorb the calories and nutrients they need to grow increasingly ludicrous muscle size. This can throw the gut bacteria out of balance and cause them to produce bloat-inducing intestinal gas. Even those who avoid having a bubble gut on stage are hardly ever able to perform a stomach vacuum as bodybuilders of the 70s and 80s routinely could, and whereas good-looking midsections were a dime-a-dozen in those times, hardly anyone today can hit a really good ab-and-thigh pose. Weight-restricted divisions such as Classic Physique have brought a return of bodybuilders with narrow midsections because they don't need to force-feed themselves and inject insulin in order to meet crazy standards of mass, and because they know that the judging criteria in their shows are more weighted towards having good conditioning and lines.
Fat itself can actually be helpful in a fight. Body fat, like muscle, helps absorb blows and can protect the body's internal organs as well as bones. Roman gladiators in particular are thought to have cultivated a layer of fat by eating barley so that their shallow, slashing blows could draw audience-pleasing blood without serious injury. (Though it does have the disadvantage of cutting down endurance, particularly if you must run about; it's still weight.)
Many strongmen and open class weightlifters seek to gain as much body mass as possible, since higher body weight gives you more stability and leverage for static lifts such as the deadlift. In a perfect world they would like to gain all of that weight in muscle, since muscle weighs more than fat and directly increases strength. However, even with the help of Anabolic steroids one will eventually reach a point of diminishing returns, where the body can hardly be persuaded to produce any more muscle tissue. Fat tissue also adds to body mass and is much easier to gain, so its only natural that they would seek to gain muscle and fat at the same time.
The physical ideal of having broad shoulders and a narrow waist comes from classical Greek artwork. In other cultures, such as the Japanese, overall strength was represented by a barrel torso, with samurai frequently depicted sporting large bellies. The idea is that a strong, powerful body needs to be firmly rooted in strong legs and hips/stomach. Which build is best for you depends on what you're trying to do. The classic V-shaped look is often and with good reason called a "swimmer's build", and is generally the ideal for sports where speed, grace and endurance are required. Athletes that focus on lifting or pushing, such as sumo wrestlers, linebackers and shot-putters, will invariably prefer the "stout" physique.