The reason ovaries develop cysts is because something is off with your hormones. It's not unusual for tubal ligation to cause such hormonal disturbances, because the blood supply to the ovaries can be affected. Without adequate blood supply, the ovaries start to atropy and just can't function the way they are supposed to. Symptoms related to this are part of Post Tubal Ligation Syndrome:
Symptoms of Post Tubal Syndrome
The synthetic hormones in the Pill do not correct this hormone imbalance, they only cover it up for a time. Once you go off, the underlying issue is still there and may have worsened. It can make the transition through menopause less gradual and more difficult to manage, the need for Hormone Replacement, higher chance of requiring a hysterectomy for heavy bleeding since the ovaries can't regulate that as well as they normally would have, etc.
Many women find that a tubal reversal actually restores blood flow and reverses the symptoms of Post Tubal Ligation Syndrome. There is a doctor in Carolina who does this.
Symptoms of Post Tubal Syndrome
Other women need Hormone Replacement to manage. This helps to manage the the symptoms and does come with it's own risks, but doesn't cure the problem.