The most obvious thing is it keeps your oven free; a range has only one oven, but four or so burners. Its also often easier to check on it, add ingredients, etc. stove-stop. Stove-top also lets you quickly turn up or down the heat. Stirring is easy stove-top, more annoying in the oven.
For what you're doing, the biggest difference is going to be where the heat comes from. Stovetop, of course, comes almost entirely from the bottom of the pot, the lid for example can remain fairly cool, meaning moisture evaporating can condense on it, and drip back in. In the oven, the heat surrounds the pot, so the lid will be just as hot—or hotter, even—than the base of the pot.
To take a recipe designed for stovetop, you'd perform all the browning steps stovetop (of course). Maybe add a little more liquid. Then bring it to a simmer stovetop, and finally transfer to the oven.
Putting a piece of aluminum foil over the pot (under the lid) can help keep moisture in.