Some general differences:
Continuous Lighting (Hot Light)
- Usually less expensive
- You can use the reflective light meter in your camera & get pretty decent exposure
- Consume a lot of electricity over time
- Generates a lot of heat which can make your model uncomfortable. Also a fire hazard if you are not careful
- If you shoot with daylight-balanced film, color correcting filter is usually needed because of the warm cast that the light produces
Strobe Lighting
- Usually more expensive
- Watt for watt, more powerful compared to hot lights
- A handheld incident light meter & a sync cord are required to get the correct exposure
- Could be a little difficult to get the lighting setup you want, especially if you don't really know about lighting (e.g. ratios, positioning, how differerent accessories change the quality of light, etc.)