Wicca is an attempt to recreate the worship of European peoples in pre- Christian times. As you may have already noticed, there are a lot of ways to honor the Old Gods. Some honor Celtic deities, others Greek and Roman, still others Norse, Italian, Slavic... the list goes on and on. There are even those who worship deities from lots of different pantheons, and still others who call only on the archetypal names of the Lord and Lady.
Asatru, on the other hand, is made up of people who worship only the old gods of the Norse and Teutonic peoples. Asatruar don't deny or show disrespect to other gods out there, they just don't worship those not a part of the Norse/Teutonic cosmology. Needless to say, professing (sort of like a Wiccan initiation) yourself into the Asatru way is not something to be done on a whim. I've been at researching this for three years as of this writing, and I still haven't done it, as I know this is not to be considered a frivolous undertaking. In Wicca, you can walk away if you decide the path is not for you. In Christianity, you can just stop going to church. In Asatru, however, once you're in, you're in it for life.
Another difference between Asatru and Wicca is the amount of magic involved. Read any good work on Asatru, and you'll have a heck of a time finding even a simple love spell (I haven't found one yet in any book of Asatru, really). In Wicca, magic is often considered an intrinsic part of the belief system. This isn't to say that there isn't any magic in Asatru; more than a few Asatruar, in fact, work with runes for just such a purpose. You won't find too many Asatruar working with anything except the runes, as objects such as Tarot cards are not native to the Norse and Teutonic peoples.
You'll know if an author is worth his or her salt if, when describing the Norse deities, they mention that there is no goddess of the moon; rather, Sunna is of the sun and Her brother Manni, of the moon. It has gotten to the point that I will stop reading any work if the author attempts to say otherwise. There aren't any Triple Goddesses or Gods in Asatru either. Each deity has their own attributes, some of which may or may not overlap with another and that is that (they are, however, grouped into the Aesir and Vanir, the former typically having more to do with functions of the sky, while the Vanir are generally more closely connected to the land).
About the Hatemonger Angle...
Many misinformed people think Asatru is something neo- Nazis get into and claim as their "religion." While we do have those hate mongers claiming to be a part of Asatru, in truth the vast majority of Asatruar don't care what your genetics gave you for skin color or ethnicity. If you're serious about seeking and worshipping the gods of the Old Norse and Teutonic peoples, there's no reason why you can't. Period.
Raven Online is a good online source for reading up on rituals and deities of this pantheon.