Amazon's Instant Video store (on the Web site) has all the major studios and networks just like Apple does. Even so, availability of content is a mixed and mysterious bag, as it is with most online video stores these days. Certain titles appear on iTunes but not on Amazon, and vice verse. Meanwhile, TV content on Amazon and iTunes has normalized, for the most part. Most TV shows are available on both services. The bottom line is, both services offer plenty of movies and shows, but have holes in their back catalogs; check through both to see what's available and missing.
Pricing might be the biggest reason to try Amazon Instant Video. So far as we know, TV and movie studios set the prices on their content, so most Apple and Amazon pricing will be identical. But Amazon seems to offer lower prices on many titles, presumably eating the difference as an inducement to get more business. Amazon also only offers movies in HD, so there's a one-size-fits-all price for rentals or sales. Rentals range from 99 cents to $3.99, and movie purchases tend to range from $9.99 to $14.99, but you'll occasionally see discounts down to as low as $4.99. Most TV episodes cost the same as they do on iTunes, but prices for the rest of the content are either lower or match iTunes prices.
If you have Amazon Prime, you get a Netflix streaming-like subscription that offers up a package of free streaming movie and TV content for customers of Amazon's $79-a-year Prime service (which also entitles you to two-day delivery of other Amazon goods with free shipping). The amount of "free" Prime content isn't as large as what you'll find on Netflix, for example, but there's more content than you might think. Currently, Prime content is, at best, a subset of Netflix's offering, even though Amazon has ramped up some exclusives at the time of launch. But there's no telling what the offerings will be like as time goes on.
Overall, Amazon tends to offer less expensive movie rentals and purchases than iTunes, but the content is hit or miss, so you'll need to experiment a bit to see if your shows and movies are available. It's unfortunate that you need to go to the Amazon Web site to purchase or rent content (why not have it in the app?), but to save a few extra bucks it might be worth it. If you're an Amazon Prime subscriber, you get even more free streaming content, but it's not on the level of Netflix Instant.