Studio lights plug into the wall, speed lights are portable. Each has their advantage:
Studio: more powerful, faster recycling time, most have modeling lights, don't have to worry about batteries
Speedlights: portable, smaller, most allow eTTL (most studio lights do not), most have high speed sync (most studio lights do not), can be used on camera and off
If you're purely going to be staying in a studio the easy choice is studio lights. If you're going to move around a lot I think speedlights are a better choice, but maybe get mobile with studio lights.
Studio lights are more powerful, but I never lack power indoors unless I'm competing with sunlight through large windows. Speedlights are plenty strong enough for portraiture and whatnot. Funny that the place they really lack is outdoor portraiture, but taking studio lights outdoors has it's own share of difficulties. I also find them lacking when doing indoor architectural work and I want to balance my light against big sunny windows. For that I use a studio strobe supplemented with speedlights - but that's a very specific use.
One of the main advantages to speedlights is eTTL and HSS. But I'm a speedlight junkie and I hardly ever use these functions. I prefer to shoot in full manual, though some really like eTTL even in portraiture.
As I said, both have their pros and cons. I admit to being biased towards speedlights, so take my recommendations with a grain of salt. I think speedlights are more flexible, allowing on-camera use, off-camera use on location, and can still hold their own in the studio. Others need high power, or ultra-fast recycling (glamour portraiture), or just hate batteries. To each their own.
