The difference is the level of speech. "Nous" is formal while "on" is more "familiar". They mean the same in some contexts. For example : nous sommes allés au cinéma / on est allé au cinéma has the same meaning. Nous sommes contents / on est content. However, there is a slight difference. "Nous" refers to identified people. On the contrary, "on" can be used when people have not been identified. An example : on a frappé à la porte (you don't know who knocked at the door), on te demande au téléphone (you are asked at the telephone). "On" is also used in idioms. Example : Plus on est de fous, plus on rit (etc...) I hope I made it clear but I'm sure other natives or French speakers will complete my post.

The previous answer by Cilla is very thorough. The only tip I would add is that on top of being a level of speech difference, it's also a distinction of medium: spoken French (on) and written French (nous). When speaking, people will simply use "on" instead of "nous" to say the same thing. You will find "nous" in most written texts unless it is the unidentified "on" that Cilla referred to.