The difference is storing nuclear waste is accepted practice today expected from every country having a plant, storing chemical emissions(CO2) is not.
A enforced storage/cleanup would be ideal yes, U.S. has never agreed to any standards, even know while Kerry talking about shutting down Coal by 2030 from nowhere, the U.S. government explicitly did not join the pledge to shutdown coal in 2030s like some countries did last week in COP26, and has always refused to get into any international binding agreement.
From a economic perspective what you are mandated to pay for [1] today is how you model costs of the project. Running a nuclear plant today means you have to keep spent in fuel on premise with no horizon for that status quo to change. Economic model for actual money going to be spent now ( not environmental or social costs models of indirect costs) has to factor that in the cost of ownership of a plant.
Is that unfair because fossil fuels has indirect costs ? yes it is, but that does not matter from an economic decision making point of view when financing a new power plant today. Carbon tax is not a solution either as currently being envisioned [1]
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[1] The carbon tax that is being discussed in the U.S. would be a disaster .
a) The tax rate is quite low which will supposedly increase over the years. The political pendulum in the U.S. almost guarantees that when republicans come to power in 2024 or later they are going repeal/relax a carbon tax like with Paris Agreement. U.S. is not currently in position to make serious long term commitments on any policy.
b) The polluters want limits *relaxed* as part of the tax deal. That means they want to be able to pollute as much as they want and just pay a small tax to do so ,which is why it is actually supported by some republicans I suppose.
c) Finally there is no plans on how to use the money to *remove* CO2 from the atmosphere, reducing emissions with that money with green investments is not good enough as polluters are in theory paying the government to take care of the problem and will pollute as they wish so government needs to clean the CO2 up. CCS is not viable economically today certainly not at the tax rate being proposed.