The advantages of studio strobes over speedlights for off-camera lighting is not as much like getting a dSLR over a P&S, as it is like getting full frame over crop. While the advantages are there and undeniable, speedlights may actually be sufficient to your usage, especially in these post-Strobist days as a lot of speedlight-specific gear is hitting the market. You have to figure out on your own when/if you need to make the move to the bigger lights. For some folks, it'll be obvious, for others less so.
Pros of Studio Strobes
As you say, this is the biggest and most obvious advantage of a studio strobe vs. a speedlight. The more power you have, the more situations you can shoot in. Situations where a lone speedlight would be outgunned by studio strobes would be groups shots (the larger the number of people in the group, the happier you'll be with a larger light), and shooting day for night in bright sunlight. Think of the power output of your light the way you'd think of maximum aperture on a lens--the more you have, the more versatile the light becomes. Just pay attention to where the low end of the range on the flash is, too and how close you like to work. The minimum power setting available on a high powered strobe may still be too much light when working in small spaces or at close distances. Likewise, a super powerful strobe if you shoot macro product photography may be overkill.
Traditionally, studio strobes are powered from a wall plug. The difference between that and using AA batteries is like water delivery with a hose vs. small buckets. You don't have to wait for a capacitor to fill up with sufficient charge for a burst. These things are built to take repeated high power pops (built in cooling/heat dissipation), so no waiting around for a flash to cool down before you can use it again. You aren't worried about your batteries going flat. You aren't managing dozens of AAs and multiple rechargers. If you do choose to use a battery pack to take a studio strobe on location, it's more like having a water tank than buckets.
Having such a reliable even flow of power, rather than the up-and-down waves from batteries and a capacitor also gives you more consistent power (i.e., speedlights can get weaker as the batteries empty), and the color remains more consistent shot-to-shot.
You can replace the bulbs
Not only is the strobe built for harder usage than a speedlight, it's also expected that your light will outlast your bulb(s). Studio strobes generally allow for you to replace a burnt bulb without a massive amount of disassembly and needing a soldering iron. Speedlights, not so much. And you can replace not just the flash tube, but also the bulbs for the modeling lights.
Speedlights attempt to fake having a modeling light with a lot of pulses of the main flash tube. Continuous it ain't, and it's mostly useless while it really runs the risk of overheating/damaging your speedlight. Studio strobes have actual continuous lights that can approximate the result of the strobe's output while you're setting up. Basically you can see what you're doing without needing to chimp, adjust, and reshoot over and over. Just move the light, see what happens. No strain to the main flash bulb. Seeing what you're doing is a good thing. And if you have a live subject standing in as you're setting up, it's far less annoying to them than multiple strobe pops.
Buit-in Optical Slaving
Studio strobes tend to have a "dumb" optical slave as a built-in feature. With speedlights, this isn't always a given. You don't have to buy radio triggers or sync cables to start using a strobe. Granted, a lot of the Strobist-style cheapies come with this feature these days, but the high-end OEM flashes from Canon don't.
Made for Modifiers and vice versa
Studio strobes have actual mount rings on them for modifiers. No need for umbrella swivels or adapters to use umbrellas and softboxes and reflectors and beauty dishes. And the array and variety of modifiers is a lot larger than for speedlights, because a studio strobe can basically just be the bare bulb. You can choose the reflector dish that goes behind the bulb, as well as the modifiers that go in front of it. With speedlights, nearly all of this stuff is jerry-rigged in some way, and almost none of it connects directly (and stably) to the flash itself. With speedlights, it's hard to get the light to point to the center of an umbrella. Not a problem for a studio strobe.
Of course you have to consider the type of mount ring, often called a 'speed ring', that's on the strobe. There are several different 'systems' (Bowens, Paul C. Buff/PCB/Alien Bees), Elinchrom, Einstein, etc). You might need to determine which system has the modifiers you want to use the most. Many modifiers from quality manufacturers are available in different mount rings (kind of like third party lenses) and can be swapped over from one type of speed ring to another using adapter rings, but the adapter rings tend to cost as much as a budget level speedlight!
Better Character of Light
Studio strobes are bigger. The character of the light they give, even bare, reflects this. Speedlights are smaller, more concentrated and the falloff is more abrupt. When both are used with a larger modifier, a larger light can fill that modifier more evenly. However, used with a smaller modifier, you may not see any difference. This difference is subtle, but recognizable. Think of it,
in some ways, as the difference between shooting full frame and crop. Crop is often sufficient for your needs and can be indistinguishable from full frame. But if you see and need that difference, then only full frame will do.
Midway Point: Bare-Bulb Flashes
Say, however, that you're used to shooting with speedlights. You just want a little more power, a little more even spread on the light, better juice to the heads, but you don't want to lug a studio strobe out onto the beach cliffs. This is where the barebulb flashes may come in. They're sort of the midway point in terms of light output, power behavior, cost, and eveness of spread between speed lights and studio strobes. They're only slightly larger than speedlights (not including the external battery pack), can be used on the hotshoe, and have a lot of the same types of modifiers as studio strobes.
Cons of Studio Strobes
Studio strobes cost more. While a low end-end studio strobe can cost less than a high-end OEM speedlight, a high-end one can cost like a full-frame body. And a super-low-end speedlight can be supercheap. In addition, when you start out as a newb, the chances are you may already have a speedlight hanging around to learn with, while it's doubtful you already happen to have a studio strobe in your gear bag.
Speedlights rock on being portable. Studio strobes are harder to lug about, and as bigger and heavier gear, the support gear (stands, booms, etc.) will also need to be bigger and heavier (and probably more expensive). Speedlights are made for batteries while studio strobes are happiest with a wall plug nearby. And because they're bigger, you may be more limited in how you can place them. There are little nooks, corners, or odd placements you can easily tuck a speedlight into that may not take a studio strobe.
Exposed flash bulb
The flip side of having a replaceable bulb is that that bulb is exposed most of the time. It can be more easily damaged in transport. You have to make sure you're protecting that bulb or have extras on hand. Speedlights have their bulbs tucked away and well-protected so throwing one in a bag isn't quite as fraught with possibilities.
No hotshoe triggering
Lower cost studio strobes typically don't have any other slaves than "dumb" optical slaves, and require cables to connect any other kind of triggering system--including radio slaves. Today, a lot of lower-end radio triggers come with hotshoe connections that only speedlights can take advantage of. So your tradeoff about worrying about the condition of your AAs is that you're worried about the condition (and connections) of your trigger cables. And, of course, studio strobes typically can't be dual-purposed for on-camera event shooting, and look a little silly if used that way [see Quadra, Profoto B2].
Lower cost studio strobes are manual-only. If you're used to shooting with a hotshoe flash, then no TTL, no HSS, no menu control, no "smart" slaving in CLS or wireless eTTL. I know, you're a tough (cheap) Strobist that does everything in Manual and this doesn't scare you. But maybe you're a YN-622 or Odin shooter who's used to all these features and convenience and it'll bug you to be able to dial in remote power on the two speedlights you're using for background and rim, but then have to walk up to your key to set the power.
Remote Control Restrictions
Remote power control is also less likely with a budget studio strobe. At best, you'll have to use a same-brand trigger that works with built-in power control on the strobe--which won't control the power on your speedlights.
What Mo' Money Buys You
As with anything, more money can buy you a more upscale label. Yes, you can peel the cartoon bee off the back of an AlienBee if it bugs your clients, but maybe it's more impressive to have Profoto on the lights. And the fit and finish, color consistency, power output, and features can go up the scale as well. Today, there are studio strobes that can do TTL and HSS, and have built-in radio triggers that will integrate with those features and grant you full remote control with your speedlights as well, without having to stack triggers (see Phottix Indra and Odins). An Einstein gets you better output and color consistency than an Alien Bee. And, of course, there's always warranty, reputation, build quality, and resale value to consider if you're thinking of cheaping out with an eBay special.