When determining your studio lighting there are a multitude of lighting options to choose from. Usually the first consideration is your budget and the cost of these options can range anywhere from $100 to $2,000 (and I'm sure there are more expensive options) for a single light or light kit. So, for me anyway, there are only three options: Strobes, Continuous lights, or Speed Lights. A lot of photographers, me included, struggle with which will work the best for them. As usual, I'm going to tell you that in my opinion it depends on what you are shooting and what you are comfortable with. This post will focus on what I see as the primary advantages and disadvantages I have experienced with each.
Strobes are what most photographers use in a studio setting. I say this because it's what I have experienced. The biggest advantage to strobes is the output of light. Different manufacturers and different models of strobes have varying output, but as a type of studio light they have far better output than either continuous lights or speed lights. What this allows you to do is shoot at varying apertures and ISO settings by changing the light settings to output an amount to match the required exposure. With this in mind your creativity from low key to high key portraits has a much wider range than you can get with the other two types of lights. The primary disadvantage of strobes is the sync speed of your camera. If you're not familiar with your sync speed take a look at your camera manual to find out what the sync speed is for your particular model of camera body. Sync speed is the fastest shutter speed your camera can shoot while using strobes. In most cameras it is either 1/250 or 1/350. If you try to shoot at a faster shutter speed it will result in a part, or all, of your image being filled with black. That's because the shutter curtain has closed before the duration of the light has completed the exposure. However, I don't really see this as much of a disadvantage for my studio shooting. You are usually concerned with shutter speed because of motion of the subject and in a studio environment you are probably not worried about stopping motion. The exception would be if you are shooting a subject like an athlete or cheerleader doing an activity that is too fast for the sync speed. This isn't an issue for me but if it were that's where the use continuous lights or speed lights would help.

Shot with Strobes using clam shell lighting with a beauty dish
Continuous lights are great for "what you see is what you get". The makes it easier to shape the light the way you want it because you can see subtle changes in positioning of the lights or the subject. This is especially great for those less experienced with studio lights. Most photographers have generally steered away from continuous lights in the past because they tended to get so hot. In fact, that's where the term "hot lights" came from. The heat would literally heat up the subject if they were close enough. Particularly bad for food photography and models if you got too close. However, in the past several years there have been big advances and these lights are fairly cool now. In my last shoot using these types of lights I was able to handle the bulbs just minutes after they had been on for about an hour. Another major advantage to using continuous lights is that you are not restricted by sync speed. Therefore, you can shoot with whatever shutter speed you desire that will result in a good exposure. The major disadvantage to continuous lights is the lack of output. I have a
TD6 by Westcott and while I love the soft light it gives I must put it very close to the subject or I will not have enough light for a good shutter speed and aperture setting. In my case, the aperture can be no smaller than F5.6 with an ISO of no less than 600 to maintain a fast enough shutter for a portrait. Even with my camera on a tripod. Of course, if I had brighter studio lighting or additional continuous lights that probably would be less of an issue, but for a one light setup in my studio it's a problem. Therefore, for me this light has become basically a fill light for my strobes.
Colton Senior 2016Colton Senior 2016
Shot with Continuous light using an eye lighter reflector
Finally, what about speed lights? You usually don't think about speed lights for your studio because like continuous lights they don't have the power that you can get with strobes. As a rule, I don't usually use speed lights in the studio but they certainly can provide a good solution as long as you know how to take advantage of their strengths. The primary strength of speed lights is their portability. If you shoot primarily on location they can be a great tool. Especially since you can get brackets that allow you to use several of them together. This setup allows you to shoot as if they were a single light source. Another advantage of using speed lights is the "Through The Lens", or TTL function you can use. This basically lets the speed lights determine the amount of output required to light the subject appropriately. The problem many photographers have with this is that it takes control away from them and gives it to the computer inside the lights. Proponents will say however, that you can always use your flash compensation to add or take away light from what the computer determines. Finally, one thing you should consider is how you will use modifiers with the speed lights. Modifiers such as softboxes and umbrellas soften the light before it falls on your subject. Using these with speed lights may be a challenge due to the amount of output required. Softening the light also removes the amount of light that will reach the subject. Due to this you may need multiple speed lights on a bracket, or move the lights close to the subject, or both. Speed lights can cost you anywhere from $100 to about $1,000 each. So if you need multiples you should consider the additional cost.
Jenna Geyman - Rockwall Shoot - 03-29-14
Shot on location with speed lights
To sum it up you have to first consider how you are going to use the lights. Strobes have great output but they're not really portable. They also have the sync speed issue, and if you need to use them for shooting moving objects it could be a problem. Continuous lights have relatively low output, but they're great for being able to shape the lighting for your shot because you can see it before you fire the shutter release. And there is not a sync speed issue to worry about. They can be a real asset if you have great lighting in your studio or don't mind moving the light very close to your subject. Finally, speed lights are very portable and great if you shoot more on location than in your studio, or if you don't have a studio. They're also great if you're OK with using the TTL function to allow them to determine the amount of light required for a good exposure. The downside of speed lights is the output. Like continuous lights they lack the amount of power that strobes have. This means you may have to use multiples together or move them close to your subject. Especially if you are using modifiers.
Hope this helps you determine what works best for you. As you can see each has their own set of advantages and disadvantages.
Happy shooting.