Inflammation would probably be caused by an infection - gingivitis, parodontitis, etc. Irritation would be caused by an allergy maybe, or maybe acidic food, or too much salt. Also, it should be: My gums are inflamed and they hurt. My gums are irritated and they hurt.

It does mean the same, essentially as we would understand irritated means inflamed, but perhaps less severe. We would also say hurt rather than hurts.