As for the definition of 'important,' tonsils are like bouncers at the party that is your body - they keep an eye out for troublemakers and do their best to toss out any pathogens that try to crash the scene. Tonsils help activate the immune system and condition white blood cells for pathogens in the environment.
If you want an actual study, here is one example:
>Conclusion: Overall, according to these findings, CD10 as a marker of B lymphocytes in children undergoing tonsillectomy was significantly less than those healthy children. This may indicate a decrease in B cells and further reduced antibody production in these patients.
I'm not sure I can begin to answer that question given the immune system's complexity... How many percent smarter does your frontal lobe's top left corner make you?
If you have to leave in most of an organ for it to function, then it's a lot more important than an organ where you can chop out the whole thing with minimal consequences.