Unispins are actually a part of freestyle. =p
Anyways, for me, Freestyle is not soo much about the tricks, but more about the performance. The music, the clothing (costumes and stuff) and the presentation. Of course, tricks do come in play. Look at the 10 freestyle lvls, and watch some freestyle videos, like this one, and youll get a good idea of what freestyle is, and you can kinda see that its more artfully done, and its like a dance.
Street kinda combined freestyle with trials, and people started taking tricks to objects, off objects, and over objects. Street has really grown a lot, and now there are tons of technical flips and spins. This video is a great example of street. He doesnt do as many flip tricks as a lot of people, but its very creative, and shows some great street riding, as does all his videos.
Now trials, its like an obstacle course. You have point a, and point b, and obstacles in between. Rocks, logs, whatever, but its up to you to navigate and jump across each obstacle to clear the line. Its my favorite style of riding, and excites me the most. Why ride across the bridge when I can hop across the rocks in the river? Its fun. This video is a good example of trials riding (right click-save as)
The difference between the unis? Well, for street and trials, there isnt much. The main thing is some people like a shorter crank length for street, around 125mm, and for trials, people like a longer crank length, around 135-145mm. Street riders also like a smoother rider, so the fad going around right now is to shave their tires.
For freestyle, the unis use a smaller tire with a higher pressure, shorter cranks, generally a higher seat, and all around a slimmer uni. Just check out unicycle.com and look at their free style unis and their trials unis and youll see the difference.