Technically, the only difference is how the word is used, like "I am obviously ignorant of that." v's "My ignorance in that subject is obvious."

Here is one for you though:

Ignorance (being ignorant):  Not knowing something because you simply have never been taught or given the information.

Arrogance (being arrogant):  believing you are better than everyone else.

Naivity (being Naive):  gullible enough that even if you have been given the right information, you are quick to believe someone when they try to con you into believing what you know is wrong.

Stupidity (being stupid):  You've been given the information, it's been drilled into your head for your entire life, there is no medical reason why you can't learn it, and the information just won't sink in, and you keep repeating the same mistakes over and over, and just don't want to know.

Shes right,but to put it simply ignorant is when you don't know anything at all about the topic or subject and ignorance is when you choose not to find out anything on a subject ,topic everyday life things . I even heard that to plead ignorance is against the law now you cannot use it in court anymore as you must have access to some form of knowledge around you e.g You commit an offence but maybe didn'y do the whole thing but you can't plead ignorance because you were their. I will look up the  meaning of it through the local court and let you know.

I decided to look up the definitions after trying to figure out how best to answer this question.

Ignorant is defined as: 

1.  lacking in knowledge or training, unlearned (an ignorant man)
2.  lacking in knowledge or information as to a particular subject or fact (ignorant of quantum physics)
3.  uninformed, unaware
4.  due to or showing lack of knowledge or training (an ignorant statement)

Ignorance is defined as:

1.  the state or fact of being ignorant; lack of knowledge, learning, information etc;

So there really is no difference between the two other than grammar for example.

He made that statement out of ignorance.  or He made that statement out of ignorant.  (Which sounds better?)

You can find these definitions just like I did at dictionary.reference.com/

Great question...made me have to use brain cells.  I guess I have gotten so used to a limited vocabulary consisting of  "No", "what did I tell you" and "get down from there" lol.  Have a great day.