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?
Update
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.
Update 2
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.
Best Answer
I think the difference is in degree. It's similar to the difference between large/huge/gigantic or small/tiny/miniscule. In some contexts, you may be able to substitute any of those words, but due to common usage or word origin or whatever other factors, there is an understood subtle scale.
So, if you have relevant facts to discuss, they may or not be important, but they are related to the matter at hand. But if you have pertinent facts to discuss, they have precise or logical relevance to the discussion. They absolutely should not be overlooked. Pertinent facts are always relevant, but relevant facts are not necessarily pertinent.
One reason why this may be the case is that pertinent could also be used as a synonym for apt (ie. strikingly appropriate) but relevant doesn't really have that meaning. If I said you wrote an "apt answer" to this question how would you feel? What if I said it was a "pertinent answer?" Or "relevant answer?"
I think most people would feel that calling it a relevant answer is nearly a mild insult. Oh yeah, it's related, but not close to perfect.