The dictionary defines relevant as being
Closely connected or appropriate to the matter at hand
whilst pertinent is defined as
Relevant or applicable to a particular matter.
Both of these definitions seem to me to be the same, and the dictionary also defines pertinence and relevance as synonyms. However I feel that in my normal usage I apply a further meaning (something I can't quite articulate) to pertinence.
For example, I see a difference between a relevant question and a pertinent one. Perhaps it's that I see a pertinent question as one which maybe alights on some previously unexplored aspect of the matter at hand, whereas a relevant question explores more familiar ground. Almost as if pertinence requires more insight than relevance.
Does anyone else feel that there is a difference between the two?
So, we're agreed that there is a difference, but I'm still not definitely clear on what that difference is. I think there's maybe more to it than Guffa's suggestion that frequency of use causes pertinence to feel more important. I feel that Robusto might be on to something with impertinence being definitely not the same as irrelevance. I will look into this more when I get the time, but in the mean time I appreciate your help.
I've accepted Ghoppe's answer, because I think that one sums up what I'm feeling the best. However (as pointed out by Noldorin in the question comments) I appreciate there's not any real difference at all in their meaning.