Besides improving my flexibility, are there other advantages to stretching?

Yes, there certainly are. Stretching is extremely relaxing and most athletes use stretching exercises to maintain a balance in body mechanics.
Combined with correct breathing techniques (Refer to Link: “Breathing Naturally” and “Nose Breathing”), it will give you a new sense of well-being and alertness. It will help you learn more about your body which will help you improve your posture.
You will find yourself doing a few stretches at work or at school just because you know it will help you release the tension and make you relax.
Most important, stretching regularly just make you feel so good!