Well as I see it both introverts and extroverts have challenges in and of themselves, it isn't specific to either one or the other.
Generally, what you find is a strength in one, turns out to be a weakness in the other, for example, lets take conversation, seeing as it is one of the most prominent difference between the two.
An extrovert generally has no problem striking up a conversation with another person, but a issue that most extroverts have is over talking (this is not strictly an extrovert issue, everyone over talks to some extent or another), they like the spotlight and feel the need to entertain their audience, and to do so they enter into long winded, flashy stories thinking that their audience is sitting on the edge of their seats with excitement when, in reality, the reason they are sitting on the edge of their seats is so that when the extrovert blinks they can run away.
Lets now look at the introverts conversation style. What some might see as a weakness can in some cases be seen as a strength. When a introvert actually DOES get involved in a conversation they are generally expert listeners and are intent on hearing and analyzing what the other person has to say.
While others may see an introverts quietness as being shy(in some cases this is the reason), what is happening most of the time is they are listening intently and gathering information, they only speak when they feel they have something important to contribute. This is a blessing as well as a curse, because while it is nice to be listened to, it is very hard to carry on a one sided conversation, you might as well be talking to a wall which to some extroverts, that may be what talking to a introvert seems like.
What I have found is that while some people may look at an extrovert from an outside perspective and see it as a burden,that it just isn't true.
Both introverts and extroverts have their challenges, and being one or the other isn't the issue, it is how you overcome the supposed weaknesses of your personality type and learn to make them your strengths.
Whichever personality type you are, know that you have an important role to play, because without the extroverts our the population would die off and everything would be boring, and without the introverts there would be no order and nothing would get done.