Speedlights or Strobes? Which one should I use?

Youtube has a good number of videos illustrating the benefits of both speedlights and strobes. Enough to make your decision as easy as pie and clear as mud.  I'm not going to bother making yet another video. That would probably only contribute to the murky and muddy waters.  I'll let you decide what works best for your type of shooting and budget.

What I will do here is help to divide the two types of lights even further apart, so as to assist you in clarifying which would be better for your particular situation.

Here goes.

Speedlights, or hotshoe flashes. Used on the camera body or off have a number of benefits. The first is that they are in general much smaller than a studio strobe.  I would also venture to say that they are considerable less expensive, but not always.
Another benefit is that they are flexible in that their small size allows them to be mounted, hidden, tucked away in interesting places in order to illuminate the scene as desired without too much trouble.  For example I have taken to wrapping my SB900 with an industrial twist-tie and attaching it to a tree branch when required. Not easy to do with a studio strobe.

Speedlights run on batteries, and therefore don't require a cord to be plugged into a wall outlet or generator. Speedlights have a plethora of accessories including colour gels, diffusors, light shapers, reflectors and more. They can easily be mounted on a low-weight light stand and coupled with a number of different light modifiers. They can be ganged together in infinite groups to create a wall or tree of light if necessary. All triggered by one trigger. If one fires, they all do. Just make sure you have enough batteries.   I love my speedlights. I use both the Nikon SB600 and SB900 almost every time I shoot portraits or still-life images.

Talking about the studio flash, or flashes that do not have a hotshoe mount, such the speedlights do means talking about larger lighting units that put out considerably more power than speedlights. In that circle we have brands such as Elinchrom, Profoto, Speedotron, Balcar, Bowens and others.
They do have a number of benefits to the studio and street photographer. 
Benefits such as: high power output, fast recycle time, some have modeling lights at more.

Some models are strictly battery powered while others require a cable connected to a power source in order to operate. Profoto for example have two lines of flash, one of which is a self-contained battery powered flash unit and another that is AC powered.
All of these lights have a huge variety of lighting accessories. Some brands less expensive than others.
Most of these brands have long pedigrees, such as Elinchrom and Speedotron.
I was a happy Speedotron user for a number of years, but I jumped off of that bandwagon when I came to the conclusion that the work I was doing didn't justify carrying the heavy power pack that always relied on a power outlet.  In fact I think the catalyst was when on a shoot for a woodworking magazine, and the electricity went out for half the city. We had to reschedule. Needless to say, I quickly secured a couple speedlights for the next day's shoot, just in case.

Studio lights have a lot going for them over speedlights. They also have a few draw backs.
First and foremost, they are powerful. If you need to light a large group, studio lights are probably your best bet. They will cover the group with power to spare.
On the negative side of the argument, they can be heavy and take up much more space. Not great if you have to travel any distance or carry your gear. They also tend to be rather pricey. Looking at a nice Profoto 2 light kit? Plan on spending between $2000 and $3500. Of course if you know that you will need that power, it might be worth the investment.  Counter that with a simple 2 light speedlight kit at about $600 to $900, and you find yourself truly looking at your lighting needs very closely.

When it comes down to lighting, it is a personal decision based on finances, intended use and the style of lighting. Weigh all of these factors together to come up with your ideal lighting kit.
You might also keep in mind that rental houses are usually available to fill out that requirement in a pinch, but remember to keep your credit card usable.