They're very close. Old Icelandic and Old Norse were once the same language. Old Icelandic was simply a dialect which after undergoing some changes has survived to this day.
The changes are mostly orthographical and phonological. Icelandic preserves a lot of Old Norse vocabulary, but there is, of course, a whole modern vocabulary to be learned when it comes to Icelandic. The orthographical changes can be easily adapted to: hestr->hestur and maðr->maður, for example. Unstressed dental fricatives in Old Norse become voiced in Modern Icelandic (gengit-> gengið, at-> að), Modern Icelandic breaks up most consonant clusters (as reflected in the orthography), and Old Norse orthography can vary quite a bit as there was no standard. Compare some Old Norse texts on sagadb.org to their Modern Icelandic versions.
Phonologically, there have been huge developments over the past six or seven hundred years. Dipthongs and a few letters are also pronounced differently in Modern Icelandic from what we know of reconstructed Old Norse pronunciation. This now begs the question, will you be learning Old Norse using a reconstructed pronunciation, or Modern Icelandic pronunciation? I'd imagine this would make a difference when it finally comes to studying Icelandic.
There are also differences when it comes to oblique case marking and so on, but don't worry about those just yet.
Icelandic is a dynamic, modern language!
Enjoy Old Norse! I also see you're a native Danish speaker, which might help you out a tiny bit when it comes to deciphering vocabulary.