Buckle in for a long answer.

There are three primary advantages that "studio" flash have over hotshoe flashes. The first, and most obvious is power; even the lower-powered "serious" units (we're not talking about AC-powered lightbulb-shaped slaves) tend to start at at least the equivalent of 2 "full-sized" speedlights (of the Nikon SB-910/Canon 600EX-RT/Yongnuo 560 class, which will weigh in at around 40-50 joules, or the equivalent thereof, depending on the zoom and dispersion settings of the built-in reflector, etc.) at about 100 joules (or watt-seconds), and of course we go considerably higher.

The second is duty cycle. That's primarily down to cooling. Even the monster units, like the Godox/StreakLight/Bolt bare bulb units, and the similar units from Quantum, lack active cooling or, for that matter, large heat sinks. Studio lights almost all have active fan cooling, and even those that don't (mostly hobbyist cheapies) will have significantly better passive cooling than flashes designed primarily to mount on a camera or bracket.

The third is modelling lights. Seeing what you're doing while you're doing it — where the shadows are falling, catchlights, reflections from eyeglasses or jewellery — is actually a good thing from time to time.

Apart from the duty cycle, the advantages for studio portraiture in the digital age, at least in the current digital age, aren't nearly what they were in the film era. For one thing, you can use considerably higher ISO settings on any reasonably-current digital camera than you would have dared to load as (colour) film, particularly if you are imagining printed output larger than 8x10 or 8x12. ISO 400 (colour) film in a 35mm camera was entirely inadequate, and a 16x20 from ISO 400 was iffy with a 6x7 (unless you were pushing the film and selling the effect). If you were shooting for publication, you'd probably be shooting chromes at ISO 25 to 100; for prints, you'd probably be shooting Vericolor III (or later, Portra) at 100 or 125 (it was a little thin at 160). These days, if you want to use a ginormous modifier with eleventy layers of diffusion in front of it with a more-or-less static subject, you can crank it up to 400 without batting an eye, and there are many cameras that don't really penalize ISO 1600 (unless you're addicted to pixel peeping). For formal(ish) portraits in a small space, it can often be difficult to get some monoblock (or pack-and-head) systems down to a low enough power these days: smaller formats mean larger apertures even if we disregard the shallow DoF trend, and ISOs lower than 100 (and sometimes 200) are scarce as hens' teeth on affordable cameras. (The D810 will go there, as will CCD MF backs, but everything else marks it as a "special Low" setting, which means "I'm going to let you overexpose here, try not to clip anything important".)

There is also a school of thought that says that chimping means the modelling lights don't matter anymore. I tend to disagree, but then I prefer my subjects to be a little freer (and a little less prepared for the click) and watch the shadows and highlights as they happen. But that's a shooting style preference now. It took a week (or, if you had deep pockets, several hours) to chimp in Ye Olde Dayes. It's certainly possible to work without the modelling lights, but if you're not actively posing your subjects, your keeper ratio goes down. But that may not matter much anymore either, since deleted bits don't cost you enough to bother with. (So you have to replace a memory card a month sooner. So what? It's not like wasting three or four rolls of souped film.)

There's no way around the duty cycle, though. You can hang some pretty heavy-duty batteries off of a speedlight (or use one of the lithium-powered units), and gang flashes to further reduce recycle times, but eventually your flash is going to either melt down or go into thermal protection mode if rapid firing is the order of the day. And it can be, depending on who you're shooting and how you're trying to portray them. Kids can eat up a lot of frames quickly, as can athletes and dancers when they're being shot as athletes and dancers.

But there can be very good reasons for having more power than a hotshoe flash can provide. Now, you can always go all Joe McNally on the scene and use a dozen or so small flashes, but it's often easier and cheaper to use studio lights. More often it's just easier.

In this question, for instance, the "target" D&G shot used a huge main light, probably a Broncolor Para 333 or the similarly-sized Profoto Giant 300 from a distance of about 4.5 meters/15 feet to get a certain combination of softness, directionality and minimal fall-off. You can probably fake most of the effect using a 7-foot parabolic umbrella, or even a wall bounce or a large silk (or, say, a king-sized bedsheet clamped to a couple of light stands playing the part of a silk), but it still needs to be 12-15 feet away from your subject and you still need to fill the modifier with light and get that light to your subject. And that's for a static subject; you'd need to pull it back further to allow your subject to move in the full frame without changing the exposure significantly, and you need more power to keep a moving subject within the zone of acceptable sharpness. A barebulb 360, or even a small gang of one-piece speedlights can probably do the job, but you'll be running at the ragged edge of full power, long recycling and overheating. You'd be right in thinking that perhaps that's not always a relevant situation in a home or small studio setting, but flying lights through a window and standing down a hallway to shoot is a thing too. (Never underestimate what you can do in a small or awkward space if you're determined to get the shot despite constraints.) A 500 joule (or better) head that recycles in two seconds or less can come in awfully handy. Two 3200J packs that can feed a single twin head in alternating sequence with a single trigger and keep up with a D4 or a 1DX at 500 joules can be even handier sometimes. (You'll know when that "sometimes" is when you run into it.)

Honestly, a hobbyist can do an awful lot these days without going anywhere near the pro gear. A lot of what drives the hobbyist into the pro market is just a bad case of GAS or some psychological need to be "just as good as the pros" (like the quality of the pictures you take can't do that by itself). A half-dozen YN560III/IV units (I've picked the cheapest fully remote-controllable units I know of, not endorsing Yongnuo in particular) and a controller, along with a relatively current enthusiast-or-better camera, a lens that's up to the task, and some relatively inexpensive stands, brackets and modifiers is a better, more versatile setup than most mid-level pros would have had at their disposal a decade and a half ago, and with a little thought, practice and effort can allow you to take pictures that would have been a real production effort at the tail end of the 20th. It may take you a little bit longer to get the shot in the can is all.

But there are still times when time is important, and the set-up and tear-down and it-just-fits-togetherness of studio gear usually beats the heck out of speedlight and speedlight accessories. Sometimes there's no substitute for space when making a shot, and with it the flash power that space demands. Sometimes you actually do need high power and fast recycling at the same time. It depends what you're shooting and why. It depends on whether or not you need to, or have plans to, make money at the game (a common though unfortunate affliction). If you can get 95% of the way there with 50% extra time spent, you can save thousands of dollars and probably wind up feeling happier and more fulfilled. If you're under the gun and that final 5% means you lose to another photographer the next time the client asks for bids, the situation is a little different. (And you won't be buying anything "just 'cuz"; if you don't actually need it, you won't get it... most of the time. The gear is unquestionably seductive. Your accountant usually isn't.)