Beyond thathey, they're your ankles! One school of thought goes that NO boot, within reason, is going to protect you if you really start to roll that ankle.
Still, a heavier bootwith its thicker leather and stiffer shankcertainly offers more support and keeps feet and ankles more stable. At a minimum, I'd recommend something such as Asolo's FSN 95 ($155; www.asolo.com) or L.L. Bean's Cresta Leather Hiker ($169; www.llbean.com). Both are substantial enough for backpacking, but won't feel like concrete overboots on a day hike.
What I'd really suggest, though, is a boot with some heft: for example, Chaco Garvins ($210; www.chacousa.com); Dunham Waffle Stomper Premiers ($180; www.nbwebexpress.com); or Lowa Trekkers ($200; www.lowaboots.com). All these boots have a nice blend of support and flexibility.