Are the two used completely interchangeably or are there specific situations where one might be used more than the other? For example I've never heard 日本語 said にっぽんご.
[EDIT] I only 'heard' that にっぽん was overused during WWII to encourage patriotic feelings. After Jesse Good comment I tried findind document backing up this but as I didn't found anything it's probably better not to remember this point !
Also I remembered that Japan was officialy read as にっぽん according to the Government, I haven't had time to check it yet but this link linsting words who are mainly read as にほん or にっぽん tends to say the two form are reckognize by the government.
Finally, I find Tim answer better than my original answer, so I adwise you to scroll down if it still appear after mine.
From what I heared, 日本 was read にっぽん but was 'overused' during WWII to suggest patriotism toward the country. In order to take distance with this meaning, people switch to a softer reading which is にほん.
Still, 日本 is still 'officially' read as にっぽん, but from what I observed people will mostly prefer the softer version except if you want to explicitly affirm your attachment toward the country, some simple example could be a footbal match, some really 'official' context (I'm mostly thinking of politicians, such as the right orientied parties you can find on saturday in Shibuya).
So I think you will mostly never be wrong using にほん in normal cases. And if you find yourself wanting to cheer for Japan (as a team) feel free to use にっぽん if you feel like it.
There is probably other situation to use にっぽん but I prefer to base me reply on situation I actually saw happen.
I think it helps to understand why we have two ways to say 日本、Nihon and Nippon. Nippon comes from the south of Japan and was used by the Choshu-Satsuma alliance (薩長同盟（さっちょうどうめい)）that overthrew the Tokugawa shogunate (aka the 幕府 (ばくふ)) in the mid-19 century. We can still see this influence on the institutions that came out the the Meiji Restoration such as the Bank of Japan which uses Nippon Ginko on its bank notes. The alliance led the modernisation of Japan in the Meiji period ultimately acquiring an empire through military expansion and establishing the country as one of the so-called "Great Powers" that dominated the world politics of 100 years ago.
Hence, Nippon tends to be used in nationalistic/patriotic situations such as charging in to battle and firing up the crowd at a political rally. It is a only small (harmless) extention to hear it being used at a football match or any other situation where the intention is to encourage patriotism.