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.
Temperature range. Beer yeasts are roughly categorized into lager and ale strains. All lager strains work best at low (~500 F.) temperatures, while ale strains work best closer to room temperature (~680 F.). But within either of these categories, different yeasts have different optimum temperature ranges. For example, WYeast 1024 (American Ale) has a maximum temperature of 720 F. while WYeast 3724 can ferment at temperatures up to 950 F.
Attenuation. Yeast strains vary in how much of the available sugars in the wort they can ferment. Ale yeast generally have apparent attenuation numbers around 75%. Lager strains are typically higher, around 80%. But again, there's a lot of variability within these groups. WYeast 3711 (French Saison), an ale yeast, can reach 83%.
Flocculation. Some strains of yeast tend to clump together during fermentation. These strains are described as highly flocculent. Flocculation and attenuation tend to be inversely covariant. That is, as flocculation increases, attenuation drops and vice-versa. This is due to highly flocculent yeast dropping out of suspension, and fermentation stalling early.
Alcohol Tolerance. Alcohol is a poison to all yeast, but some strains are more tolerant than others. Most yeast will tolerate up 9% ABV, but some, particularly Belgian strains, go as high as 13%
Some yeast traits are objective and quantifiable in principal, but in practice they are subjective in the absence of laboratory equipment;
Ester Production. Yeast produce small quantities of esters, which give beer a taste and aroma usually described as fruity.
Phenol Production. Similar to esters, yeast pro due phenols which are described as giving beer a spicy quality.
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.
What would happen if I used a different strain than the kit recommended?
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:
pay attention to temperature guidelines. In particular warm fermentation with lager yeast will usually produce a beer wreaking of sulphur. Some lager strains (California common, Bohemiam pilsner) can be fermented warm but they're the exception not the rule.
if you're brewing a high-acohol beer be sure to use an alcohol tolerant strain unless you like your beer sweet.
low alcohol beers (ordinary bitter, mild, etc) work best with a high flocculation, low attenuation strain. These strains leave enough unfermented sugars in the beer to provide some body and sweetness.
Some beers get the bulk of their flavour from the yeast strain. Wheat beers and Belgian ales are good examples. You can experiment with different yeasts when making these beers, but you'll probably want to stay within the same type of yeast strain. I.e. don't try to make a Belgian tripel with an Irish ale yeast.
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.
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.