One thing before we tackle the problem:
The one thing you should never try to do (but you might want to do) is try to change them. Your SO's personality is there to stay- remember, that's why you fell in love with them in the first place- and to try and change such a huge part of it is just not worth the effort. They also will get the sense that you don't love them for who they are, and that can take even the strongest of relationships down.
What you can do, though, is try to alternate activities. Plan date nights, and every other time, have them choose the outing or activity that you guys do. Maybe you want to go out dancing one night, and he wants to play board games and drink a bottle of wine the next date night. You can enjoy a whole host of different activities. Having both an introverted and extroverted person in one relationship can be stressful, but it can also make for a very adventurous partnership. The extrovert would not normally have stayed at home on a Friday to have a picnic on the patio and look at the stars, but they probably will love it anyway. The introvert probably wouldn't have chosen to go to a bar to see a new music act that has been playing in town, but they might find a new favorite band because of it.
Being different shouldn't be an issue, as long as you compromise. In fact, it may even be for the better.