In alpine conditions generally skis will work better. Below timberline skis work on broad trails. Narrow twisty trail and bushwhacking is easier with snowshoes.
Snowshoes on a steep crusted side hill are double plus ungood.
Note that snow conditions have a strong effect on both. In Eastern Canada snow is often wet and heavy. I have snowshoed there when I would sink only 2-3 inches, and going without snowshoes was mid thigh.
In western Canada's front ranges (eastern ranges of the rockies) snow tends to be very light and fluffy, and the first person on snowshoes would still go in knee deep even on 60 x 13" snowshoes.
On the prairies in Manitoba, snow often comes sideways, and is packed hard by the wind. Snowshoes then become traction devices on a crust strong enough to bear your weight, and smooth enough to land you on your butt several times an hour.
(I was talking with a Cree trapper once. He was all modern with a polaris dual track snowmobile, but he said that his grandfather had a collection of snowshoes for different conditions and jobs. The ones for ice fishing on Lake Winnipeg had no lacing on toe and tail, and only very coarse lacing -- about 5/8" wide on 2.5" spacing on the centres. This was enough to keep traction on the crusts, and on the ice that was swept bare, and also to bridge cracks in the ice.)
If you are using traditional wax skis temperature has a huge effect, particularly temps within a degree or two of freezing. This can get really annoying if the trail is alternately in sun and shade, as finding a wax combination that works acceptably for both conditions may be difficult to impossible. This isn't as big a deal if you are going mainly uphill with skins, then mainly downhill without.
Snowshoeing solo is a lot of work breaking trail. The second person is spending less than half the energy.
Snowshoes leave a trail about 18" wide: perfect for a pulk or other sled.
Both can be worth wearing even in a few inches of snow, just for their terrain averaging. 8" of snow hides a lot of small logs, rocks, gopher mounds, and trail ruts. A snowshoe or ski bridges enough of these, that your foot strike is more uniform, and you aren't as sore at the end of the day. even those short 24" aluminum and neoprene are effective for this.
There is a lot of variation in both kinds of equipment. Snowshoes range from cheap plastic, metal tube and synthetic lacing, traditional wood frame and rawhide lacing. Sizes range from 18" to 6 feet, with widths from 10 inches to 14 inches. Snowshoes can either have tails (track better, tips don't get caught under the crust as often) or round ends (easier in bush.
Skis can be very narrow (~50 mm -- fast on broken trail) or wider (~100 mm -- more common on back country) There are skis that attempt the compromise between snowshoe and skii being about 8" wide by 5 feet long. The width makes them hard to edge, but combined with short length makes them more usable in heavy brush.
Overall: Conditions favouring skis:
- Open terrain
- Steep slopes
- Side hills greater than 10%
- Windpacked or crusted snow.
- No brush.
- Experienced personnel.
Conditions favouring snowshoes:
- Deep soft snow
- Irregular terrain
- Lots of brush or trees.
- Newbies. (It takes about 10 minutes to learn how to use snowshoes)