Stern versus Nordhaus is most importantly a disagreement about the discount rate.
But an underappreciated argument is offered by Martin Weitzman who argues that instead of viewing investment now for the future within the context of cost-benefit analysis, we should do so in terms of insurance. What we are doing is paying now to offset the chances of a worst case scenario, one that may have a low probability of occurring but would have catastrophic circumstances if it did. (See “The Stern Review on the Economics of Climate Change,” Journal of Economic Literature 45 (2207): 703-724.) But is there really a conceptual conflict here or not? First, an insurance policy is the wrong metaphor. Insurance does not prevent worst case outcomes, it distributes some measure of offsetting compensation over those who suffer such outcomes and those that do not. So insurance assumes an uneven distribution. Worst case climate would affect us all. Moreover our interest is in costs to prevent such outcomes not to offset their costs to those affected. If that is right, the issue is not a matter of cost-benefit analysis versus some alternative but rather how much weight to give to the worst case scenarios even if their probability is small.