"Wiggle room" is informal. It's spoken language. You use it in writing only when you want your writing to sound informal. "Leeway" is neutral. It's neither formal nor informal. It is originally (and still is) a nautical term. The lee side of a ship is the side away from the wind. Imagine a ship sailing north, with the land on the west, and wind blowing from the east. The wind is blowing the ship toward the land, a dangerous situation. The left side of the ship is the "leeward" side. The land is "on the ship's lee." "Leeway" is the amount of distance the ship has on its lee side. A ship with more "leeway" is safer. It has more room to maneuver, more room to steer. Therefore, "leeway" means "a safe space within which you can maneuver." Indeed, instead of "leeway" you can say "room to maneuver." "Wiggle room" suggests that you are caught in a tight space. You might be trapped. However, by wiggling, you might be able to wiggle your way forward and get out of it. The more "wiggle room" you have, the better your chances.

Leeway is definitely more formal than wiggle room. An even better choice in formal settings might be "flexibility." Hope this helps!