hi! not discouraging anyone from supplying a direct response, but FYI, there are several posts on the general topic of Norse conversion in the FAQ* - check them out!

Norse: end of the Viking Age and conversion to Christianity

*see the link on the sidebar or the wiki tab

I'm not sure what to add to what others have said. I really like this book for a general introduction to Vikings, and it covers a bit of their early conversion to Christianity. http://www.amazon.com/The-Vikings-History-Robert-Ferguson/dp/0143118013

If you want more advanced stuff you cannot go wrong with anything written by Sverre Bagge. He taught Norwegian history for many years at the university of Bergen (I'm not sure if he still does) and is one of the leading scholars on Viking history and has an entire book on their conversion.

I wouldn't say there were any real benefits to Vikings becoming Christian other than just being included more in terms of diplomacy with Christian controlled Europe. A theory out there suggests the Viking age may have actually began as a reaction to the Christian expansion of Charlemagne. Norwegians (I'm not as familiar with other Scandinavians) actually fought for centuries to reject the church, especially in the areas around Trondelag. There were Christians in Norway but they were just a small and slowly growing community. Haakon the Good was the first real major figure to try and convert the nation but he failed for the most part. Harold Bluetooth (the name should sound familiar) was the next to make the push and had more success, but Haakon "the bad" (as he is remembered for being against Christianity) was able to shake off the Danish king Bluetooth and return the country to paganism. Except for some, who revolted and resulted in Haakon being killed by a servant.

Olaf Tryggvason made the third major, and possibly most influential, impact on converting Norway. He did so with a good amount of violence. How much is legend and how much is fact is somewhat blurred but many agree Olaf persecuted non Christians in Norway pretty heavily, and sometimes tragically in the case of the pagans. His rule was short (5 years) but what he did worked in a lot of ways. Many Norwegian kings after him were Christian, such as Olaf Haraldsson (aka St. Olaf), and over time Christianity became much more dominant within the country.

So it wasn’t exactly something that happened naturally and wasn’t accepted by the majority for centuries.