One of the unfortunate things about Iceland is that you can't just camp anywhere. Most of the time you have to camp in campgrounds of some kind. If you pull over somewhere in Iceland and you think it's not owned by someone, you apparently would be very wrong. It seems that every bit of the island is owned by someone. Either someone who lives there or by the national park itself. I'm also assuming that some of the lands just can't be lived on due to it being too cold and the possible snow that could occur. Unfortunately, the biggest difference between Iceland and New Zealand is that they don't have an app that shows you where all the campsites are so you really have to guess on locations. 

The biggest recommendation I can give for that sort of thing is to download the whole country of Iceland's google maps on your phone so that you have something to use for when you don't have any service. Most of the time its accurate as to where you need to go and where the campsites are so if you are worried about getting lost or not having a good guide while you are on the road then check that out. 

Just make sure to not randomly pull over while you are in the country or you could get a little visit from someone who will not be a happy camper or even worse the nice little coppers will give you a visit.  I'm sure you don't want either of those when you are over in a country that you don't know very well. I will say that getting visits from either would probably be a rude awakening after a long day of wandering, hiking and etc. 

So please download the entire google maps of Iceland before you go because there really aren't any good apps out there (that I could find anyway) that will give you the information that you need to know when it comes to camping in Iceland