I have only brewed ~6 batches of beer thus far, and for all of them I have used Wyeast smack packs. However, for each batch, the kit I was using recommended a different strain of Wyeast. What is the difference between each strain of yeast, and what would happen if I used a different strain than the kit recommended?
There are a number of objective, quantifiable differences between yeast strains.
Some yeast traits are objective and quantifiable in principal, but in practice they are subjective in the absence of laboratory equipment;
Different strains are also described as producing beers that are "malty", or "minerally". Some strains are noted for producing "tart" beers, but this might just be a side-effect of unusually high attenuation accentuating acids already present in the wort.
To complicate matters further, many of these traits change with fermentation temperature. Belgian yeast strains will typically produce phenolic beers when fermented at cooler temperatures, and beer heavy in esters when fermented warmer.
You would make a different beer. Sometimes the difference will not be particularly noticeable. You could substitute Scottish ale yeast for American ale yeast, and your beer would be subtly different. But if you used Belgian Abbey Ale yeast instead of American ale, you would produce a noticeably different beer.
Experimenting with different yeast strains is one of the most rewarding parts of home brewing, in my opinion. Not all of your experiments will be roaring successes, but it's hard to make a beer completely undrinkable just by varying the yeast strain.
A few guidelines to help you in your experiments:
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:
While an alternative saison strain (Wyeast 3711) is described:
(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.
I just wanted to add a point. WIth liquid yeast, it's important to use a starter. Not only will you save money on buying 2 liquid yeast packs but it will ensure your fermentation gets off to a good start. Certain styles you want the yeast to be stressed (like Weizens) because it gives off a banana or clove flavor. However with most styles this isn't desired.