Whether or not your boots are crampon compatible is a big deal. If you are going to be in glacier country and occasionally doing some light glacier trekking, then you may want a stiff hiking boot that is crampon compatible. These are distinguished by the crampon groove around the toe of the boot. However, there are a type of crampon (strap on) that don't require a groove to fit. If you anticipate more time on the ice, you may want a mountaineering boot. All mountaineering boots are crampon compatible; and because they are stiffer, you are able to do some low-angle ice climbing in them.
Mountaineering boots have a much stiffer shank than hiking boots. Hiking boots are easily bent as you walk over terrain, but mountaineering boots are stiff enough that they can support ice climbing with crampons. Overall mountaineering boots will take a heavier beating.
Both mountaineering and hiking boots can offer good ankle support, though, generally, mountaineering boots will provide a higher and stiffer support. All models are slightly different, and it depends on how hard you want to trek.
Most mountaineering boots are going to be heavier than your average hiking boot. They are stiffer, bigger and simply contain more materials (some are made of plastic). There are some pretty lightweight mountaineering boots on the market, however, like the La Sportiva Nepal EVO GTX, which weighs in at a mere 1 lb. 12 oz. to just over 2 lbs. depending on your shoe size.
If you are planning on spending any nights out in winter conditions, then your feet may get pretty cold. Hiking boots usually have just a thin layer of insulation, if any at all, whereas mountaineering boots are much warmer. Some mountaineering boots even have removable liners to keep you extra toasty.