Not all mechanical keyboards are loud or clicky. I use one with Cherry Browns and mostly type without bottoming out the keys, and it's no louder than cheap keyboards.
Ugh, it's so much noise for your neighbor though. I worked next to someone with an old IBM keyboard for a year and was grumpy the entire time. Couldn't find headphones strong enough to cancel out what sounded like someone shooting an automatic weapon in the office.
I think there's a happy medium with the quieter cherry switches. But yeah, I don't use a mechanical keyboard in settings where I'm right next to other people.