Can tooth cavities and the fillings cause permanent damage to your teeth?

Yes. Cavities have destroyed part of the tooth by definition. Fixing the cavity removes the bad part of the tooth and a little more to ensure you are touching "good" tooth. So you have a hole in the natural tooth that is filled or restored with different materials depending on how big the cavity was.
Yes. Cavities left untreated can cause permanent damage to teeth to the extent of needed them removed. On the other hand if they are treated with fillings or other restorations like inlays or crowns cavities can and should be eliminated from the teeth with minor damage.
In this way... Once your teeth are decayed and require fillings, they are not quite as strong and healthy as what we call "virgin" teeth. Nothing is as good as what nature gives you with healthy teeth. So, yes you can say they have been permanently damaged. Practice good oral hygiene, eat healthy, and see your dentist to prevent cavities, or treat them early if "damaged" by decay.