Tonsils, in my experience, are quite common to have removed for various reasons (tonsillectomy). The most common one is to abate bad breath caused by those who suffer with tonsil stones. Other reasons could be viral, inflammation etc... The tonsils are seen as largely benign so it would have no longterm effect on the patient. (This is the belief, of course*). It serves a purpose in the immune system, but people don't want poo poo breath, therefor this is a common and easy procedure.
Now the appendix serves as part of the lymphatic system and helps the body by keeping the digestive system functioning at tiptop shape after diarrhea and all that jazz. Since it sits right at the most caudal point of the large intestine, it may run the risk of being infected with stool and other bacteria if its opening gets clogged due to abnormal expansion of it's opening. Now this is all just guess work as science still can't 100% decide on its function. Appendicitis is usually the reason behind appendectomies.
Fun fact! The appendix (and other organs, teeth, bones etc...) are considered vestigial organs. So these are organs that are believed to at some point in our evolution to have had an important purpose. Not so much anymore. As our diets and lifestyles change overtime, other organs will serve bigger purposes and others will wither to the side too.
