Its no shock to me that many never understand the difference between been brilliant and been intelligent, because it took a while for myself to understand the concept behind those words. Its kind of hard to give a full explanation on the difference between the two words, but will be giving a glimpse of insight about them.
Technically brilliant people tend to remember, understand and assimilate when been thought. Doesn't really matter either fast assimilation or a slow one, as far as he/she is able to understand and put down in full details what he/she has learned.
All this are also applicable to intelligent people except, they love to think outside the box, formulate their own methods and most times are good crammers.
For example, in a lecture room, the teacher explain some points and give notes on it. During a test, a brilliant student who has read well write exactly what is in the note and an intelligent student who might not go through the note except when the teacher explains it in class will find other suitable words and align them. If its noticed sometimes, we might notice some of our colleagues don't read at all and still score higher than others that burn candles in the night. The intelligent ones even attend parties the time others are reading.
The Indian movie "3 idiots" is a perfect example. Silencer who was the most brilliant student according to the movie was always second to the intelligent playful one. He never read or stress himself more than usual and still the overall best student.
All intelligent people are brilliant and might be easy for brilliant people to be intelligent if they can only think outside the box of what they've been thought.