Yeast strains have many different properties, primarily related to the types of flavor and aroma compounds they produce, the effect of fermentation temperatures, pitch rates and re-pitching, alcohol tolerance, flocculation and even what types of sugars they can consume.
For instance, the "Dupont" saison strain (Wyeast 3724) is described:
A traditional yeast that is spicy with complex aromatics, including bubble gum. It is very tart and dry on the palate with a mild fruitiness. Expect a crisp, mildly acidic finish that will benefit from elevated fermentation temperatures. This strain is notorious for a rapid and vigorous start to fermentation, only to stick around 1.035 S.G. Fermentation will finish, given time and warm temperatures. Warm fermentation temperatures at least 90°F (32°C) or the use of a secondary strain can accelerate attenuation.
While an alternative saison strain (Wyeast 3711) is described:
A very versatile strain that produces Saison or farmhouse style biers as well as other Belgian style beers that are highly aromatic (estery), peppery, spicy and citrusy. This strain enhances the use of spices and aroma hops, and is extremely attenuative but leaves an unexpected silky and rich mouthfeel. This strain can also be used to re-start stuck fermentations or in high gravity beers.
(When it says 3711 is extremely attenuative, it's not kidding: I've had beers ferment down to 1.002 at not particularly elevated fermentation temps (~70°F).)
In the case of many beer styles, the types of flavors the yeast produces are considered appropriate or even required for the style. For example, Weihenstephan Weizen (WY3068) "produces the banana and clove nose traditionally associated with German wheat beers and leaves the desired cloudy look of traditional German wheat beers."
At the same time, there is a much more subtle difference between some strains, making multiple strains completely appropriate for a given style. For instance, a stout fermented with Wyeast 1272 (American Ale II/Anchor Liberty) might only differ in some subtle fruity or nutty notes compared with the relatively clean profile of Wyeast 1056 (American Ale/the "Chico" strain/Siera Nevada).
There are more major differences between ale (Saccharomyces cerevisiae) and lager (Saccharomyces pastorianus) species, and again with the various Brettanomyces species.