All things being equal, why would one arm pit sweat more than the other. And not just a little more, a lot more. Like, hypothetically speaking, if one were to wring out a t-shirt, nothing would come out of the left side, but the drippings from the right could probably fill up a bucket.
It’s arm-pit cancer, isn’t it? I just know it. Arm-pit cancer. Great. Now there’s a dignified death.
KEN WHEATON 
Maybe the glands on one arm pit are all stopped up.
Also, you are now “my gross friend”. Congratulations.
You can try ArmPit Botox!
http://www.sweatyarmpits.net/medical_procedures.html
Probably just that you use that one arm more than the other. You probably use that arm to um, shake mixed drinks or something…