Cork panels alone won't solve your neighbor problem because they absorb but don't isolate - you need to block the sound transmitting through the walls, not just dampen it inside the room. What actually works is combining mass (like drywall or dense sheets) with dead air spaces between layers, so if you've got the budget, build a floating wall with fiberglass or mineral wool insulation in the middle before closing it up.
If you can't do major work, at least cover the walls with something heavy like mass loaded vinyl under the cork - the combo of weight plus absorption gives way better results than cork alone, and it's cheaper than professional acoustic panels.