A "real" studio flash (as opposed to a hotshoe flash) will have many advantages over the continuous lights.
The two main advantages are:
1. The studio flash will have a modeling light which allows you "what you see is what you get" lighting and negates the only advantage of continuous lights: being able to "see" your light.
2. The flash will not put out enough heat to cook your subjects. You can use your studio flash with a softbox or umbrella quite close to your subject and therefore take advantage of the softer light from the softbox or umbrella up close. Note: there are continuous cool lights but they are quite expensive.
Other advantages of studio flash are:
1. Your modeling lights will not make your subjects squint and will not cause their pupils to contract looking like a dope addict.
2. You do not need to worry about shutter speed (except keeping it within the sync capabilities of your equipment). Your exposure is controlled by your f/stop.
3. It is very easy to modify studio flash using umbrellas, softboxes, snoots, barndoors, honeycomb grids, etc.; without the danger of burning the modifying material.
4. Your studio flash equipment will not get hot enough to cause burns to your subjects or to you as continuous lights can do.
5. Your studio flash equipment doesn't draw a large amount of current. It is easy to blow fuses or circuit breakers when working with several continuous lights.
6. You can trigger your studio flash with cheap eBay infrared triggers because the studio strobe is equipped with an optical slave. I have been using an eBay trigger in my studio for several years now and it has worked flawlessly.
Finally, while studio strobes are more expensive than trying to use home improvement center work lights; there are many studio strobes which are low priced as better continuous lights and the lower priced studio strobes can do an excellent job.