Goats have the advantage of eating almost anything (including your shirt if you get too near). They're very playful, but their idea of play tends to be what humans view as getting butted really hard. When they eat down a weed-infested area, they tend to leave it very uneven, and will girdle any trees they have consistent access to.

Sheep are much more difficult to lead around than goats (an area without sheep in it is the most terrifying thing ever to a sheep). They are exceedingly stupid. But they crop an area down flat. If you want an England-quality lawn, and have enough pasturage to make it work, you want sheep.

Miniature goats and sheep, though they are easier to handle than the standard varieties, take forever to mow even a small area. But if you have children, they'll adore them. Miniature angora goats are pretty much the cutest thing ever, particularly when they're kids.

Why do we use livestock instead of mechanical mowers and why don't we just get rid of the grass?

Answering the second question first, the grass is required as part of our landfill closure plan. Without it, the topsoil cover would erode during the winter rains here in California. Erosion would allow damage to the engineered clay cap that seals the top of the landfill and keeps the garbage and methane from escaping.

During the rains, the grass grows waist high, then it drys out around May, posing a real fire hazard to us and to our industrial neighbors. Of the grass removal methods we have available, mowers cause fires--goats do not. When we used tractor mowers, they twice started grass fires, and one of the fires spread to 10 acres before the fire department put it out.

Also, as Umbra pointed out, goats help to eradicate weeds. We have seen a sharp reduction in the amount of noxious yellow star thistle since we started using goats. The fertilizer they leave behind is a plus.

Finally, buried all over in the top foot or two of the landfill cover is a lot of plastic pipe (under vacuum) that collects the landfill gas and moves it to the big engine-generators where it is burned to make electricity. When we used tractors, their weight did a lot of serious, but hidden damage to those pipes, which posed real problems for the engine-generators. The last time we hired tractor mowers to cut the grass, the cost of the gas system damage was more than the dollar value of the contract, and we were chasing vacuum leaks for months.

And of course, the goats and sheep are far more pleasing than tractors. Our landfill gets a lot of recreational foot traffic and the walkers really like having the critters around.

That being said, mowers are loud, also smell bad, and don't leave fertilizer behind. Not only that, but you have to ride them over and over again, while the goats can pretty much do it on their own.

If you have very good fencing and some way to keep an eye on them, goats are the better choice IMHO.

Sheep are easier to keep than goats but require more protection from predators.  Their manure is similar to goat manure.  They will eat many of the same things as goasts but are less destructive.  If they are a woolly breed, they will require shearing.  Hair breeds do not require shearing.  A friend keeps a herd of sheep to reduce brush around his home and therefore reduce the danger from fires.  I kept a sheep for one season until the grass ran out, fed her grain for a month, and sent her to the butcher.  DELICIOUS.

I now have miniature horses.  They also do a fine job of grazing, although their waste needs to be picked up.  Horse manure is the finest-smelling manure on the planet and will make your garden very happy.  They do require more care and money than goats or sheep, and they must not be exposed to too much lush growth at one time unless you want an expensive bill from the vet to treat colic or founder.  We keep them in a pen and let them out to mow as needed.